9.5 Counting principles  (Page 3/12)

A family of five is having portraits taken. Use the Multiplication Principle to find the following.

Finding the number of permutations of n Distinct objects using a formula

For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let’s look at two common notations for permutations. If we have a set of $n$ objects and we want to choose $r$ objects from the set in order, we write $P(n,r).$ Another way to write this is $n_{}{P}_r,$ a notation commonly seen on computers and calculators. To calculate $P(n,r),$ we begin by finding $n!,$ the number of ways to line up all $n$ objects. We then divide by $\left(n-r\right)!$ to cancel out the $\left(n-r\right)$ items that we do not wish to line up.

Let’s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is $6\times 5\times 4=120.$ Using factorials, we get the same result.

There are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken 3 at a time. The general formula is as follows.

Note that the formula stills works if we are choosing all $n$ objects and placing them in order. In that case we would be dividing by $\left(n-n\right)!$ or $0!,$ which we said earlier is equal to 1. So the number of permutations of $n$ objects taken $n$ at a time is $\frac{n!}{1}$ or just $n!\text{.}$

A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.