Chapter 8.5, Problem 49E

To determine

To find: the number of distinguishable permutations of the group of letters.

Answer to Problem 49E

2520

Explanation of Solution

Given information:

Given group of letters

A,L,G,E,B,R,A

Calculation:

Let us consider the following group of letters

A, L, G, E, B, R, A

If a set of $n$ elements has $n_1$ of one kind of object, $n_2$ of 2^nd object…and so on with

  $n=n_1+n_2+\mathrm{...}+n_k$

Then number of distinguishable permutations of the $n$ object is given by

  $\frac{n!}{n_1!\cdot n_2!\cdot \cdot \cdot \cdot \cdot n_k!}$

The above sequence of letters is made of 2 A’s, 1 E, 1 G, 1 L, 1 B, 1 R.

So the distinguishable permutations are given by

  $\begin{array}{c}\frac{7!}{2!1!1!1!1!1!}=\frac{7!}{2\times 1}\\ =\frac{7\times 6\times 5\times 4\times 3\times 2\times 1}{2}\\ =7\times 6\times 5\times 4\times 3\\ =2520\end{array}$