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Chapter 11: Problem 49

In how many distinct ways can the letters of the word DALLAS be arranged?

Short Answer

Expert verified
The word 'DALLAS' can be arranged in 180 distinct ways.

Step by step solution

01

Determine the frequencies of each letter in the word

First, it is necessary to count how many times each letter appears in the word 'DALLAS'. There are 2 As, 2 Ls, one D, and one S. This gives frequencies of \( r_1=2, r_2=2, r_3=1, r_4=1 \) for A, L, D, and S respectively.
02

Calculate the factorial of the total number of letters

Next, calculate the factorial of the total number of letters. For the word 'DALLAS', this is \( n=6 \). So, \( n!=6!=6*5*4*3*2*1=720 \). This is the total number of arrangements if all letters were different.
03

Calculate the factorials of the frequencies of each letter

Calculate the factorials of the frequencies for each letter. This is \( r_1!=2!=2*1=2, r_2!=2!=2*1=2, r_3!=1!=1, r_4!=1!=1 \).
04

Divide the total number of arrangements by the factorials of the frequencies

The final step is to divide the total number of arrangements by the products of the factorials of the frequencies. So the number of distinct ways the word 'DALLAS' can be arranged is \( \frac{n!}{r_1! r_2! r_3! r_4!}= \frac{720}{2*2*1*1}=180 \).

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