Committees with conditions

Count committees with rules, by formula, and let the checker compare your answers with a full listing.

The idea. For a committee of $r$ from $m$ men and $w$ women, exactly $k$ women can be chosen in $^{w}C_{k} \times {}^{m}C_{r - k}$ ways. At least one woman means all committees minus those with none.

Your tasks.

1. In exactly, return the number of committees with exactly k women.
2. In at_least_one, return the number with at least one woman, by the complement.
3. In at_least, return the number with at least k women, by adding the cases k, k + 1, and so on up to r.
4. In words, return the number of words with v vowels and c consonants, chosen from nv vowels and nc consonants, all letters different.