3. How many strings of eight English letters are there a) that contain no vowels, if letters can be repeated? b) that contain no vowels, if letters cannot be repeated? c) that start with a vowel, if letters can be repeated? d) that start with a vowel, if letters cannot be repeated? e) that contain at least one vowel, if letters can be repeated? f) that contain exactly one vowe

First of all we know there are Twenty six (26) letters in the alphabet.

Five (5) vowels

Twenty one (21) consonants

a) that contain no vowels, if letters can be repeated?

No vowels = all consonants

Hence,

21⁸ = 37,822,859,361letters

b) that contain no vowels, if letters cannot be repeated?

Hence,

21*20*19*18*17*16*15*14 = 8,204,716,800letters

c) that start with a vowel, if letters can be repeated?

Hence,

5*26⁷ = 40,159,050,880letters

d) that start with a vowel, if letters cannot be repeated?

Hence,

5*25*24*23*22*21*20*19 = 12,113,640,000letters

e) that contain at least one vowel, if letters can be repeated?

Hence,

26⁸-21⁸ = 171,004,205,215letters

f) that contain exactly one vowel, if letters can be repeated?

Hence,

8 * (5*21⁷) = 72,043,541,640letters