Chapter 9.1, Problem 7E

Solution

To calculate: The number of 9-letter words that can be formed from the word LOGARITHM.

The resultant value is $362,880$ words of 9-letters.

Given information:

The given word is “LOGARITHM” consisting nine letters.

Formula used: The number of permutations of n objects taken r at a time denoted by $P_{n_r}$ and given by $P_{n_r}=\frac{n!}{\left(n-r\right)!}$ .

Calculation:

9 letters are to be selected from a group of 9, then

  $P_{9_9}$

Rewrite the expression using the formula $P_{n_r}=\frac{n!}{\left(n-r\right)!}$ ,

  $\begin{array}{c}{P}_{9_9}=\frac{9!}{\left(9-9\right)!}\\ =\frac{9!}{0!}\end{array}$

Now expand the factorial,

  $\frac{9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{1}=362,880$

Therefore, the value of $P_{9_9}$ is $362,880$ . It means there are $362,880$ ways in which the 9-letter words can be formed.