Answer:
1) First, let's count the total number of letters in the word ARKANSAS. There are 8 letters in total.
2) Next, let's count the number of times each letter appears:
A appears 3 times
R appears 1 time
K appears 1 time
N appears 1 time
S appears 2 times
3) If all letters were different, the number of distinguishable permutations would be 8! (factorial of 8). However, since there are repeated letters, we need to divide by the number of permutations of the repeated letters.
4) For the letter A which appears 3 times, there are 3! ways to arrange them. Similarly, for the letter S which appears 2 times, there are 2! ways to arrange them.
5) Therefore, the number of distinguishable permutations is:
[tex]$\frac{8!}{3!2!}$[/tex]
6) Let's calculate this:
[tex]$\frac{8!}{3!2!} = \frac{8 * 7 * 6 * 5!}{(3 * 2 * 1) * (2 * 1)} = \frac{8 * 7 * 6 * 120}{6 * 2} = \frac{40320}{12} = 3360$[/tex]
Thus, there are 3360 distinguishable permutations of the letters in the word ARKANSAS.
Step-by-step explanation:
