Chapter 8, Problem 85RE

To determine

To calculate:

The number of distinguishable permutation value for the following group of letters:

  $C,A,L,C,U,L,U,S$

Answer to Problem 85RE

The number of distinguishable permutation value for $C,A,L,C,U,L,U,S$ is $5040$

Explanation of Solution

Given information:

  $C,A,L,C,U,L,U,S$

Formula used:

The permutation value is expressed as:

  $P{}_r=\frac{n!}{\left(n-r\right)!}$

Calculation:

In the given group of letters,

There are total $8$ letters that means:

  $n=8$

Number of $C$ : $n_1=2$

Number of $L$ : $n_2=2$

Number of $U$ : $n_3=2$

Number of $A$ : $n_4=1$

Number of $S$ : $n_5=1$

The distinguishable permutation value for $C,A,L,C,U,L,U,S$ is calculated as:

  $\begin{array}{l}P=\frac{n!}{n_1!\cdot n_2!\cdot n_3!\cdot n_4!\cdot n_5!}\\ P=\frac{8!}{2!\cdot 2!\cdot 2!\cdot 2!\cdot 1!}\\ P=\frac{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{8}\\ P=5040\end{array}$

Therefore, the distinguishable permutation value for $C,A,L,C,U,L,U,S$ is $P=5040$