How many arrangements of the letters in the word BASHFUL can be made if the first letter must be a vowel?

To find the number of arrangements of the letters in the word "BASHFUL" where the first letter must be a vowel, we start by identifying the vowels in the word. The vowels present in "BASHFUL" are A and U.

Since the first letter must be a vowel, we have two scenarios: the arrangements starting with A or the arrangements starting with U. After choosing the first letter, we need to arrange the remaining letters.

The total number of letters in "BASHFUL" is 7, and when one vowel is fixed as the first letter, we have 6 letters left to arrange. The remaining letters are the ones left after removing the chosen vowel.

The number of arrangements of these remaining 6 letters (which are now all distinct) is calculated by taking the factorial of the number of letters:

6! = 720.

Since we can start with two different vowels (A or U), we multiply the number of arrangements for one first letter by the number of choices for the first letter:

2 (choices for the first vowel) × 720 (arrangements of the remaining letters) = 1,440.

This results in a total of 1,440 distinct arrangements of the letters in "B