Permutations and Combinations – Part 5- Grouping & Distribution

Grouping and Distribution

This is a very important concept of permutation and combination where some higher order fundamentals of permutation and combination is involved – the reason for reserving this topic for the end.

Here is the list of articles in this series of permutation-combination for quick reference. Ideally you should check these articles in order to gain better understanding of the whole concept –

  1. Principle of Counting
  2. Permutations Introduced
  3. Permutations – Special cases
  4. Combinations – Introduction
  5. Combinations – Grouping & Distribution

What is the difference between grouping and distribution?

To distribute something, first grouping is done. Only after  you have made groups of some objects, you might want to distribute these groups in various places. For example, after you made groups of some toys, you might want to distribute these groups among some children. Or, after dividing some number of toffees into groups, you might want to distribute these groups into boxes.

Just as the objects that we group can be similar or dissimilar, so can the places that we assign these groups to be similar or dissimilar.

While distributing groups, we need to keep one rule in mind: We permute the groups only if these places for distribution are dissimilar, otherwise not.

Say, we have 2 items- X and Y and I have to split them into two groups. There is only one way of doing it – X goes in one group and Y in the other. However, if I have to distribute among 2 people A and B, then these 2 groups can be permuted in 2! ways.

Division of dissimilar items into groups of EQUAL SIZE

Let’s take a very simple example. In how many ways can you divide 4 different things (say A, B, C and D) into two groups having two things each?

You would like to say that we select two things out of the four and two would be left behind, i.e. 4C2 = 6 ways. But are there really 6 ways?

Take a look. We can divide four things, A, B, C and D into two groups of two in the following ways:

AB – CD

AC – BD

AD – BC

You can keep trying but there is no fourth way to do it. So where have the remaining 3 ways calculated through 4C2 = 6 disappear? If you look carefully, there was an overlap. When we select 2 things out of 4, we can do it in 6 ways – AB, AC, AD, CD, BD and BC but when we select the first three groups, the last three get automatically selected without having to select them separately and vice-versa. So when we select AB, CD is automatically selected and vice-versa.

This overlap will manifold itself if we increase the number of items further.

So be very careful not to apply the usual combinations formula whenever we have to divide into groups of equal size. However, if the groups contain unequal number of things, then our earlier method of using combinations formula for selection will be valid as will be discussed in the next section.

 

Let me now increase the objects to 5. How would you divide these 5 distinct (dissimilar) objects into groups of 2, 2 and 1?

The single object can be chosen in 5C1 = 5 ways. The rest of the 4 objects can be divided into two equal groups in 3 ways as explained above. Therefore, total number of ways = 5 x 3 = 15.

  • The number of ways in which mn different things can be DIVIDED equally into m groups, each group containing n things = (mn)!/(n!)^m x 1/m!

  • The number of ways in which mn different things can be DISTRIBUTED equally into m groups, each group containing n things = (mn)! / (n!)^m

Note: In the distribution, order is important hence the divisible things can be arranged in m! ways since things are divided into m groups.

Division of dissimilar items into groups of UNEQUAL SIZE

Say we have k things and we have to divide them into 2 groups containing m and n things respectively such that m+n =k, then this can be done in k!/m!.n! ways. This is because m things can be selected out of k things in kCm ways and when m things are taken, n things are left to form the other group of n things which can only be done in nCn =1 way. Hence the required number of ways is kCm = m+nCm = (m+n)!/(m+n-n)!.n! = k!/m!.n!.

We can extend the same concept for increased number of groups as long as the number of items in all the groups add up to the total i.e. n.

The number of ways in which n distinct things can be DIVIDED into R unequal groups containing a1, a2, a3, ……, aR things (different number of things in each group and the groups are not distinct)

= nCa1 × (n-a1)Ca2 × … × (n-a1-a2-….-a(r-1))CaR

=n! / a1! a2! a3!…….aR!   (here a1 + a2 + a3 + … + aR = n)

 

What if there are n distinct things and we have to find out the number of ways in which they can be distributed among r persons such that some person get a1 things, another person get a2 things, . . . . and similarly someone gets aR things (each person gets different number of things)?

Number of ways in which n distinct things can be divided into R unequal groups containing a1, a2, a3, ……, aR things (different number of objects in each group and the groups are distinct)

=n! / a1! a2! a3!…….aR!x R!  (here a1 + a2 + a3 + … + aR = n)

Division of IDENTICAL / SIMILAR ITEMS into Groups

Number of ways in which n identical things can be divided into r groups, if blank groups are allowed i.e. each can receive zero or more things (here groups are numbered, i.e., distinct), where 0≤r≤n = (n+r-1)C(r-1)

 

Number of ways in which n identical things can be divided into r groups, if blank groups are not allowed i.e. each receives at least one item (here groups are numbered, i.e., distinct), where 1≤r≤n = (n-1)C(r-1)

 

Number of ways in which n identical things can be divided into r groups so that no group contains less than m items and more than k (where m<k) is coefficient of xn  in the expansion of (xm + xm+1 +…..xk)r

 

Some Important Results:

  • Number of squares in a square having ‘n’ columns and ‘n’ rows = 12 + 22 + 32 +….+ ∑n2 = n2 = n(n+1)(2n+1)/6

  • Number of rectangles in a square having ‘n’ columns and ‘n’ rows= 13 + 23 + 33 +….+ n3 = ∑n3

  • =[n(n+1)/2]2

  • Number of quadrilaterals if m parallel lines intersect n parallel lines = mC2 x nC2

 

With this discussion on Grouping and Distribution we come to an end on the topic Permutations and Combinations.

Please post in your doubts, queries and interesting problems in the comments box.

Hope you enjoyed reading these 5 articles as much as I enjoyed writing them.

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Permutations and Combinations – Part 4- Combinations

This article is the fourth in a series of 5 articles. Here we will be discussing a pet topic of all competitive examinations – Combinations. The list of 5 articles of the permutation combination series is –

  1. Principle of Counting
  2. Permutations Introduced
  3. Permutations – Special cases
  4. Combinations – Introduction
  5. Combinations – Grouping & Distribution

I would advise you to go ahead with this article only after you are clear on first 3 articles in this list above.

You will be better equipped to move on with Combinations after you are thorough with the above topics. We will discuss various cases of Grouping & Distribution in the article following this.

All right, let us get started!

Difference between Permutations and Combinations

Suppose there are 3 bags (A,B and C) in my home and I want to select any 2 out of them to take with me on my holiday. In how many ways can I make the selection?

Clearly I select either AB or BC or AC i.e. 3 ways.

An important point to note is that we are talking about selection and not order here. Obviously whether I select AB or BA makes no difference.

Let us take one more example.

Suppose from a class of 10 students I have to select 3 students for a play, it is a case of Combinations. But, if I have to arrange 3 students in a line from a class of 10 students, it is a case of Permutations.

I hope I have made it clear that in permutations (rearrangement) order matters but in combinations (selections) order does not matter.

We are now in a position to define Combinations.

 Definition of Combinations

Combinations is the selection of some or all of a total of n number of things.

If out of n things we have to select r things (1≤r≤n), then the number of combinations is denoted by  nCr =  n!/r!(n-r)!

Combinations does not deal with the arrangements of the selected things. This explains division by r! which denotes the arrangement of the selected r things.

Important relation between Permutations and Combinations

‘r’ selected things can be arranged in r! ways.

So, r! x  nCr  = nPr

or,  nCr = nPr / r!

or     nCr = n! / r! (n-r)!

Example: In a class there are 6 boys and 5 girls. In how many ways can a committee of 2 boys and 2 girls be formed?

Solution: 2 boys can be selected out of 6 in 6C2 ways.

2 girls can be selected out of 5 in 5C2 ways.

So the selection can be made in 6C2 x 5C2 ways. (Product Rule: ‘and’ stands for multiplication)

Example: In a class there are 6 boys and 5 girls. A committee of 4 is to be selected such that it contains at least 1 boy and 1 girl.

Solution: There are 3 different possibilities now-

i. 1 boy and 3 girls

ii. 2 boys and 2 girls

iii. 3 boys and 1 girl

In the 1st possibility, total number of combinations =  6C1 x 5C3

In the 2nd possibility, total number of combinations =  6C2 x 5C2

In the 3rd possibility, total number of combinations =  6C3 x 5C1

But only one of the above possibilities will occur; 1st OR 2nd OR 3rd.

So the total number of required combinations is  6C1 x 5C3 +  6C2 x 5C2 + 6C3 x 5C1

Some Important Results on Combinations

  • nCr = nCn-r                                           (0≤r ≤n)
  • nC0 = nCn = 1
  • nCr + nCr-1 = n+1Cr                  
  • If nCp = nCq , then p = q or p + q = n (p,q € W)

Restricted Combination

The number of combinations of ‘n’ different things taken ‘r’ at a time  subject to restriction that p particular things –

i) will never occur = n-pCr

ii) will always occur = n-pCr-p

Number of ways of selecting one or more things from a group of n distinct things = nC1 + nC2 + nC3 + …… +  nCn  = 2n – 1 .

Number of ways of selecting zero or more things from a group of n distinct things = nC0 + nC1 + nC2 + nC3 + …… +  nCn  = 2n

With this we conclude the topic Combinations. Feel free to post interesting questions which we all can share and solve together. In the following article we will discuss the rules on Grouping & Distribution- a very important concept of permutation and combination.

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Permutations and Combinations – Part 3- Special Cases of Permutations

There are 5 articles in this learning series of permutation-combination. Here’s the list –

  1. Principle of Counting
  2. Permutations Introduced
  3. Permutations – Special cases
  4. Combinations – Introduction
  5. Combinations – Grouping & Distribution

Since you are already familiar with the fundamental principle of counting and the concept of permutations, we can move ahead with some special cases of permutations in this article.

By now you already know:

# With respect to fundamental principle of counting, ‘and’ stands for multiplication & ‘or’ stands for addition

# nPr= n!/ (n-r)!

# Permutations of n different things taken r at a time = nPr = n! / (n-r)

# Permutations of n different things taken all at a time = nPn = n!

# Permutations of n different things taken r at a time, when one particular thing always occurs = r.(n-1)P(r-1)

# Permutations of n different things taken r at a time when a particular thing never occurs = (n-1)Pr

Let us begin with a question.

 In in how many ways can the letters of the word WATER be arranged so that we have a new pattern every time?

This is permutation of n different things taken all at a time which is equal to n!

Hence, total number of different arrangements possible is 5! =120.

Another way to look at it is we have 5 places to be occupied by 5 different letters.

The 1st place can be filled by any of the 5 letters, hence 5 ways. The 2nd place can be filled by any one of the remaining 4 letters as one letter has already been fixed at the first place, hence 4 ways. Similarly, the 3rd place can be filled in 3 ways and the 2nd in 2 ways. The 5th place can be filled in only one way as there is no choice but to fill it by the remaining 1 letter.

So going by the product rule, this can be done in 5x4x3x2x1 = 120 ways.

Permutation of n things when some are identical or Permutation of n things not all different:

What happens when we have to find out the number of permutations when certain items are identical?

If 2 exactly similar red chairs(R1 & R2) and 1 black chair(B) are to be arranged, then please note that one cannot distinguish between the 2 red chairs. This is to say that there is no difference between R1 B R2 and R2 B R1 because they will both look the same as I cannot differentiate between R1 and R2 as they are exactly same.

So the total no. of arrangements possible will be 3! / 2! = 3. They will be BRR RBR RRB.

We have divided by 2! to take care of the ‘two’ items that are same.

If out of n things, p are exactly alike of one kind, q exactly alike of second kind and r exactly alike of third kind and the rest are all different, then the number of permutations of n things taken all at a time                  = n! / (p!q!r!)

 Example: In how many ways can the letters of the word COMMITTEE can be arranged

i. using all the letters

ii. if all the vowels are together

 Solution:

i. Total letters = 9 and identical letters are 2M 2T and 2E.

So total no. of arrangements = 9! / 2!2!2!

ii. Since all vowels must appear together we consider them as one unit. There are 4 vowels- O I E E. So now we have 5 letters. Out of these we have 2M and 2T. These 5 letters can be arranged in 5! / 2! 2! ways.

In the group of 4 vowels, the 4 vowels can arrange themselves in 4!/2! ways.

So total no. of words formed = 5!/2! 2! X 4!/2!

Permutations where repetitions are allowed

While dealing with letters and digits, you will often come across cases where repetition in permutation is allowed or not allowed. You have to be very careful as to what is asked for because the treatment for both the cases is absolutely different.

Example: How many numbers of 5 digits can be formed with the digits 0,1,2,3,4

i. if the digits cannot repeat themselves

ii. if the digits can repeat themselves

 Solution:

i. The 1st place (ten-thousandth place) can assume only non-zero digits. Hence it can be occupied in 4 ways. The 2nd place can be occupied by any of the remaining 4 digits, i.e. 4 ways. Similarly, the 3rd, 4th and 5th place in 3, 2 and 1 ways respectively.

Total no. of numbers formed = 4x4x3x2x1 = 96

ii. The 1st place (ten-thousandth place) can assume only non-zero digits. Hence it can be occupied in 4 ways. Since repetition is allowed, the 2nd, 3rd, 4th and 5th places can all be filled in 5 ways each i.e. we have a choice of 5 digits (0,1,2,3,4) for each place.

So total no. of numbers formed = 4x5x5x5x5 = 4×54 =2500

The number of permutations of n different things taken r at a time, when each may be repeated any number of times in each arrangement is nr.

Geometrical arrangements

Circular permutation:

Sitting in a circle is not the same as sitting in a straight line. A circle does not have any starting point or ending point. Thus in a circular permutation, one thing is kept fixed and the others are then arranged relative to this fixed item. Then it is treated like a linear arrangement.

The number of circular permutations of n different things taken all at a time = (n-1)!

Fix any one as reference point, the remaining other n-1 things can be arranged in (n-1)! ways.

What if we are taking into consideration beaded necklace or a garland wherein clockwise and anticlockwise arrangements are the same?

We simply divide (n-1)! by 2 to take into account the two same clockwise and anticlockwise arrangements.

If the clockwise and anticlockwise orders are not distinguishable, then the number of permutations = (n-1)! / 2

Arrangement around a regular polygon:

If n people are to be arranged around a p sided regular polygon, such that each side of the polygon contains the same number of people, then the number of arrangements possible is n!/p.

For example, 15 people are to be arranged around a pentagon shaped table having 3 people on each side of the table, number of arrangements will be 15!/5.

Please note if the polygon is not regular, then the number of arrangements will be n! irrespective of the sides of the polygon.

 Special case of arrangement around a rectangular table:

Rectangle is a special case because though it is not a regular polygon, it is a symmetrical quadrilateral with opposite sides equal. So, if n people are to be arranged around a rectangular table, such that there are the same number of people on each of its 4 sides, then the total number of arrangements possible is n!/2. Here 2 signifies the degree of symmetry of the rectangle.

 

With this we have completed all the concepts of permutations. In the following two articles, we will discuss combinations and rules for grouping and distribution.Feel free to ask any doubt or question that arises in your mind after reading this article.

 

Keep the comments coming. Your feedback is very important.

Till then, happy permuting!

 

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Permutations and Combinations – Part 2- Permutations Introduced

This article is the second of a series of five articles. In the first part we had discussed the fundamental principle of counting. The first part is essential for you to understand the difference between ‘and’ & ‘or’ in the realm of permutations and combinations. We will discuss the concept of permutations in this article and permutations- special cases, combinations and Grouping & Distribution in the three articles following this.

Here is the list of articles in this series of permutation-combination for quick reference –

  1. Principle of Counting
  2. Permutations Introduced
  3. Permutations – Special cases
  4. Combinations – Introduction
  5. Combinations – Grouping & Distribution

Permutations

The arrangements of a given number of things taking some or all of them at a time are called permutations.
For example, the permutations of three alphabets x, y, z taken two at a time are xy, xz, yx, yz, zx, zy.
A point to be noted is that arrangement or order is very important in permutations. Hence xy is distinctly different from yx.

If r things are taken at a time out of a total of n things, then the total number of permutations is denoted by nPr.

nPr= n!/ (n-r)!

Now you will ask why is this so. Let’s clear this.
First object can be selected in ‘n’ ways. Second object can be selected in (n-1) ways. Third object can be selected in (n-2) ways. Similarly the rth object can be selected in (n-(r-1)) = (n-r+1) ways. Therefore the total number of ways of arranging these ‘r’ objects = n x (n-1) x (n-2) x (n-3) x ……x (n-r+1)
={n x (n-1) x (n-2) x ……(n-r+1) x (n-r) x (n-r-1) x…. 3 x 2 x 1} / {(n-r) x (n-r-1) x…..x 1}= n! / (n-r)!

Hence nPr  = n! / (n-r)!

Let us now understand further with the help of different examples.

Example 1: There are 4 boxes. Find the total number of arrangements if we can arrange only 2 boxes at a time.

Solution: Out of 4 boxes, we are arranging 2 at a time.
So total number of arrangements possible is 4P2 = 4! / (4-2)! = 4! / 2! = 4x3x2x1 / 2×1  = 12
Let us verify. Let us name the boxes A, B, C, D.
Total number of arrangements possible are AB, BC, CD, BA, BC, BD, CA, CB, CD, DA, DB, DC.

Permutations of n different things taken r at a time = nPr = n! / (n-r)!

Example 2: In the above example, what if all the 4 boxes are selected at a time? How many arrangements are possible then?

Solution: Total no. of arrangements possible = 4P4 = 4! / (4-4)! = 4! / 0! = 4! = 4x3x2x1 = 24.

Permutations of n different things taken all at a time = nPn = n!

Example 3: If out of the 4 boxes, one particular box should always be selected; then how many arrangements are possible if 3 boxes are selected at a time?

Solution: Since one box should always be selected we have to select 3-1 boxes out of 4-1 boxes. This can be done in 3P2 = 3! / (3-2)! = 3! / 1! = 3x2x1 / 1 = 6 arrangements.
With each of these 6 arrangements our preselected box can be arranged in 3×6 = 18 ways.

Wondering how?

Let us name these boxes A, B, C and D and D has to be always present.
So now A, B and C can be arranged as AB, AC, BA, BC, CA, CB.
With AB, D can be arranged as DAB, ADB, ABD i.e. 3 ways.
D can be arranged with the remaining 5 arrangements similarly.

Hence in total there can be 18 arrangements.

Permutations of n different things taken r at a time, when one particular thing always occurs = r.(n-1)P(r-1)

Example 4: How many arrangements are possible if out of the 4 boxes – A, B, C and D one particular box D is never selected, taken 2 at a time?

Solution: Since D is never to be selected, we have to take into account A, B and C.
We can arrange A, B and C taken 2 at a time in 3P2 = 3! / (3-2)! = 3! / 1! = 3x2x1 / 1 = 6 ways. i.e. AB, AC, BA, BC, CA, CB.

So when one particular item is never chosen, we just ignore it and treat the problem as if that particular item is not present in the total number of items.

Permutations of n different things taken r at a time when a particular thing never occurs = (n-1)Pr.

# Quickermaths Tip:
We know that nPr = n! / (n-r)!

Let us say we have to find out 12P3.

12P3= 12! / (12-3)! = 12! / 9! = 12x11x10x…..x1 ⁄ 9x8x7x…1 = 12x11x10 =1320

Instead of writing out so much, the moment you see 12P3 you should know that you have to multiply 3 numbers.

Starting from 12, we take in 3 numbers in the descending order and multiply them out.

Learn to get into the habit of writing 12P3 =12x11x10 straightaway.

This helps in faster calculation.

Hope the concept of permutations is clear to you by now. We shall discuss some special cases of permutations including circular permutations in the article following this after which we will move on to Combinations and Grouping & Distribution.
Keep your comments and other interesting problems coming.

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Permutations and Combinations – Part 1- Fundamental Principle of Counting

Permutations and Combinations is one of the most logical phenomenon of mathematics wherein there are no formulae to mug up. Rather, it tests your ability to understand the problem and interpret the situation logically. It is more of application of common sense. That is why you will see that most questions can be solved without actually knowing the technicals of permutations and combinations. Great news, isn’t it? 😉

Since it is an extensive topic, I am going to break this article into 4 parts (click on the links to read further)-

  1. Principle of Counting
  2. Permutations Introduced
  3. Permutations – Special cases
  4. Combinations – Introduction
  5. Combinations – Grouping & Distribution

Before proceeding further, let us quickly define Factorial.

Factorial of a number or n! is the product of n consecutive natural numbers starting from 1 to n. Factorial word is represented by ‘!’ or ‘L’. Hence, 4! is 1x2x3x4 = 24.

Note: Factorial of zero or 0!=1

Fundamental Principle of Counting

Product Rule

If one operation can be done in x ways and corresponding to each way of performing the first operation, a second operation can be performed in y ways, then the two operations together can be performed in xy ways.

If after two operations are performed in any one of the xy ways, a third operation can be done in z ways, then the three operations together can be performed in xyz ways.

Let us take an example.

permutation combination

A, B, C and D are four places and a traveller has to go from A to D via B and C.

He can go from A to B in 4 ways and corresponding to each way he can take any one of the 2 ways to reach C. Hence A to C can be reached in 4×2=8 ways.

Corresponding to each of these 8 ways of reaching C from A, there are 3 ways to reach D and the traveller can choose any one of them.

Hence, A to D can be reached in 4x2x3=24 ways!

Here the different operations are mutually inclusive. It implies that all the operations are being done in succession. In this case we use the word ‘and’ to complete all stages of operation and the meaning of ‘and’ is multiplication.

One more example before we move ahead.

A tricolour flag is to be formed having three horizontal strips of three different colours. 5 colours are available. How many differently designed flags can be prepared?

Solution: First strip can be coloured in 5 ways, second strip can be coloured in any of the remaining 4 colours, and the third strip can be coloured in any of the remaining 3 colours.

Hence, we can get 5x4x3 = 60 differently designed flags.

Addition Rule

If there are two operations such that they can be performed independently in x and y ways respectively, then either of the two jobs can be done in (x+y) ways.

Let us take the example of four places A, B, C and D taken above.

permutation combinationThere are 4 different roads from B to A and 2 different roads from B to C. In how many ways can a person go to A or C from B? The answer is 4+2=6 ways.

Here, the different operations are mutually exclusive. It implies either of the operations is chosen. in this case we use the word ‘or’ between various operations and the meaning of ‘or’ is addition.

The product rule and the addition rule signify the cases of ‘and’ & ‘or’.

Deciphering between these ‘and’ & ‘or’ is the main concern of most questions of permutations and combinations. So you have to read the questions very carefully and keep answering this question in your mind – Do I have to choose between the two jobs or perform both in succession? If you can answer this correctly which of course you will be able to with careful reading and interpretation and some practice, you have already won the battle!

Stay tuned for conceptual clarity on permutations (introduction & special cases) and combinations. Keep the comments coming, folks. 

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