Chapter 9.1, Problem 10E

Solution

To calculate: The number of distinguishable 11-letter words that can be formed from the word CHATTANOOGA.

The resultant value is $1,663,200$ words of 11-letters.

Given information:

The given word is “CHATTANOOGA” consisting eleven letters.

Formula used: If an n -set contains $n_1$ objects of first kind, $n_2$ objects of a second kind, and so on, with $n_1+n_2+\cdots +n_k=n$ , then the number of distinguishable permutations of the n -set is given by $\frac{n!}{n_1!n_2!\cdots n_k!}$ .

Calculation:

There are total 11 letters in the word “CHATTANOOGA”. There are 3 A’s, 2 T’s, and 2 O’s in this word.

The values are $n=11,n_1=3,n_2=2,\text{and}{n}_3=2$ 11,

Use the formula $\frac{n!}{n_1!n_2!\cdots n_k!}$ and substitute the values,

  $\frac{11!}{3!2!2!}$

Now expand the factorial,

  $\frac{11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3!}{3!2\cdot 2}=1,663,200$

Therefore, there are $1,663,200$ ways in which the 11-letter words can be formed.