Question

1. if there is a crossword that has five squares that need to be filled, how many possible five-letter arrangements are there (assuming they do not have to make a real word and no letter can be repeated)?

21!
\frac{26!}{5!}
26!
\frac{26!}{21!}

Explanation:

Step1: Identify the problem type

This is a permutation problem where we need to find the number of ways to arrange 5 letters from 26 (the English alphabet) without repetition. The formula for permutations of \( n \) objects taken \( r \) at a time is \( P(n,r)=\frac{n!}{(n - r)!} \). Here, \( n = 26 \) (total letters) and \( r=5 \) (letters to arrange).

Step2: Apply the permutation formula

Substitute \( n = 26 \) and \( r = 5 \) into the formula \( P(26,5)=\frac{26!}{(26 - 5)!}=\frac{26!}{21!} \).

Answer:

\(\frac{26!}{21!}\) (the last option)