This article is the fourth in a series of 5 articles. Here we will be discussing a pet topic of all competitive examinations – Combinations. I would advise you to go ahead with this article only after you are clear on:
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)
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.