The English alphabet contains $21$ consonants and $5$ vowels. How many strings of $7$ lowercase letters of the English alphabet (repeats allowed) contain:
(a) exactly one consonant
(b) exactly two vowels
(c) at least one consonant
(d) at least one vowel and at least one consonant
For (a) I got $C(21,1) \times C(7,1) \times 5^6$
For (b) I got $5^2 \times C(7,2) \times 21^5$
I have no idea if these are correct. The ones I struggled with are (c) and (d), could someone help me with the last two please?
Edit: After reading the comment by David G. Stork:
For (c), I did $26^7 - 5^7$. Since each of the $7$ positions can be one of $26$ letters, so total number of strings is $26^7$, then minus all the strings that does not have a consonant to get the strings that has at least one consonant.