Final answer:
To form 4-letter words from the alphabet with no repeated letters, we consider different cases. (a) For the 2nd letter to be a vowel, we have 5 choices for the 2nd letter and 21 choices for the remaining 3 letters. (b) For exactly one vowel, we have 4 positions for the vowel and 5 choices for the vowel and 21 choices for the remaining 3 consonants. (c) For at least one vowel, we subtract the total number of words with no vowel from the total number of words possible.
Step-by-step explanation:
To solve this problem, we need to consider the different cases separately.
(a) For the 2nd letter to be a vowel, we have 5 choices for the 2nd letter (a, e, i, o, u) and 21 choices for the remaining 3 letters, since we cannot repeat letters. So, the total number of 4-letter words with the 2nd letter being a vowel is 5 * 21 = 105.
(b) For exactly one vowel, we can have the vowel in any position (1st, 2nd, 3rd, or 4th). So, there are 4 positions where the vowel can be placed. For each position, we have 5 choices for the vowel, and 21 choices for the remaining 3 consonants. So, the total number of 4-letter words with exactly one vowel is 4 * (5 * 21) = 420.
(c) For at least one vowel, we can subtract the total number of words with no vowel from the total number of words possible. The total number of words possible is 26 * 25 * 24 * 23 (since we can repeat letters and order matters). The total number of words with no vowel would be 21 * 20 * 19 * 18 (since we can only choose from the 21 consonants and order matters). So, the total number of 4-letter words with at least one vowel is 26 * 25 * 24 * 23 - 21 * 20 * 19 * 18 = 358,848.