Question

how many 8-letter ""words"" using the 26-letter alphabet (letters can be repeated) either begin or end with a vowel?

Answer 1

There are approximately 21,876,340 8-letter "words" using the 26-letter alphabet (letters can be repeated) that either begin or end with a vowel.

Step-by-step explanation:

To calculate the total number of words, we consider two cases: words starting with a vowel and words ending with a vowel.

Words starting with a vowel:

Vowel choices: There are 5 vowels (a, e, i, o, u).

Remaining letters: For the remaining 7 positions, each can be filled with any of the 26 letters.

Total possibilities: 5 (vowels) * 26^7

Words ending with a vowel:

Remaining letters: Similar to the first case, 7 positions can be filled with any of the 26 letters.

Vowel choices: For the last position, there are 5 vowel options.

Total possibilities: 26^7 * 5 (vowels)

Total words:

Sum the possibilities from both cases:

Total words = Words starting with a vowel + Words ending with a vowel

Total words = 5 * 26^7 + 26^7 * 5

Total words ≈ 21,876,340

Therefore, there are approximately 21,876,340 8-letter "words" that either begin or end with a vowel using the 26-letter alphabet with repeated letters allowed.

Answer 2

The total number of 8-letter words that either begin or end with a vowel is 2 × (5 × 26⁷).

Case 1: Words that begin with a vowel
For the first letter, there are 5 possibilities (a, e, i, o, u). For the remaining 7 letters, each letter can be any of the 26 alphabet letters (including vowels). So, the total number of words in this case is 5 × 26⁷.

Case 2: Words that end with a vowel
Similar to Case 1, there are 5 possibilities for the last letter. Then, for the remaining 7 letters, each letter can be any of the 26 alphabet letters (including vowels). So, the total number of words in this case is also 5 × 26⁷.

Adding the results from both cases, the total number of 8-letter words that either begin or end with a vowel is 2 × (5 × 26⁷).