Chapter 13, Problem 9CT

To determine

To calculate: The number of distinct $8$-letter words (meaningful or not) can be formed from the letters in the word REDEEMED.

Answer to Problem 9CT

Solution:

The number of distinct $8$-letter words (meaningful or not) can be formed from the letters in the word REDEEMED is $840$.

Explanation of Solution

Given information:

The $8$-letter words (meaningful or not) can be formed from the letters in the word REDEEMED.

Formula used:

The number of permutations of $n$ objects of which $n_1$ are of one kind, $n_2$ are of second kind, ……, and $n_k$ are of a kth kind is given by, $\frac{n!}{n_1!\cdot n_2!\cdot \mathrm{.........}\cdot n_k!}$.

Calculation:

There are total $8$ letters in the word REDEEMED.

The letters in the word are not distinct and order matters so use the formula of permutation for non-distinct objects.

In the given word REDEEMED, there are four different letters such as R, E, D and M.

The letter R is repeated only $1$ time, E is repeated $4$ times, D is repeated $2$ times and M is repeated only $1$ time.

Hence to find number of ways use the formula,

$\frac{n!}{n_1!\cdot n_2!\cdot n_3!\cdot n_4!}$

Here, $n=8$, $n_1=1$, $n_2=4$, $n_3=2$ and $n_4=1$

Hence the above formula becomes,

$\frac{8!}{1!\cdot 4!\cdot 2!\cdot 1!}$

$\Rightarrow \frac{8\cdot 7\cdot 6\cdot 5\cdot 4!}{1!\cdot 4!\cdot 2!\cdot 1!}$

$\Rightarrow \frac{8\cdot 7\cdot 6\cdot 5}{1\cdot 2\cdot 1}$

$\Rightarrow 840$

Therefore, the number of distinct $8$-letter words (meaningful or not) that can be formed from the letters in the word REDEEMED is $840$.