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-
- Principle of Counting
- Permutations Introduced
- Permutations – Special cases
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
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.
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.
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.
There 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.