We have learned about permutations and combinations in our school. This topic particularly deals with the problem of arranging and grouping certain things, taking a particular number of things at a time. We can understand permutations and combinations through the next example. Let us suppose that a banker has the difficult task of filling two positions from a group of six equally qualified employees. Since none of the lot has had actual experience, he decides to allow each of them to work for one month in each position before he makes the decision. We need to know how long the bank can operate before positions are filled by permanent positions. This situation requires an efficient counting of the possible ways in which the desired outcome can be obtained. A listing of all possible outcomes is desirable, but it is very likely to be tedious and subject to errors. Here is where permutations and combinations will come in handy.
The ways of arranging or selecting a smaller or equal number of persons or objects from a group of persons or collection of objects with due importance being paid to the order of positioning or selection are called permutations.
Formula for calculating permutations of n different things taken all n things at a time is given by:
nPn= n (n – 1) (n – 2) (n – 3) …. (n – n + 1)
Formula for calculating permutations of n things chosen r at a time is given by
nPr = n (n – 1) (n – 2) (n – 3) … (n – r + 1)
where the product has exactly r factors.
Let us see an interesting example from permutations. We need to calculate how many three letters words can be formed using the letters of the words SQUARE.
Solution: Since the word ‘SQUARE’ contains six different letters, the number of permutations of choosing 3 letters out of six equals = 6P3 = 6 × 5 × 4 = 120.
While solving sums of permutations, we pay due regard to order. However, there are situations in which order is not important. For example, let us consider the selection of 4 players from the lot of 20 players. Here, we will not be concerned about the order in which they are selected. In this scenario, combinations will be used.
The number of ways in which a smaller or equal number of things are arranged or selected from a collection of things where the order of selection is not important is called combinations. The selection of a poker hand which is a combination of five cards selected from 52 cards is a beautiful example of a combination of 5 things out of 52 things.
Let nCrdenote the required number of combinations. Then the formula of combinations is as follows:
nCr= n! / r! ( n – r )!
Let us see an interesting example to understand the calculation of combinations. A team is to be formed of 3 persons out of 12. We will have to find a number of ways of forming such a team.
Solution: We want to find out the number of combinations of 12 things taken 3 at a time which is given by:
12C3 = 12! / 3! (12 – 3)! [ from the given formula above]
= 12! / 3! 9! = 12 × 11 × 10 / 3 × 2 = 220
Thus, the number of ways in which such a team can be formed is 220 ways.
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