How many 6 letter words can be made in English with at most 2 vowels? How many 6 letter words can be  made in English with at most 2 vowels?
 A: Divide the problem and solve it. First the English alphabet have 5 vowels and 21 consonant. Now back to the solution, we are checking special cases:
Words containing now vowels
Because there are 21 consonants and we need to make $6$ letter word the total number is:
$$21^{6} \text{words containing now vowels}$$
Words containing 1 vowel
In this case we have one place for vowel and we have 5 options, and for the consonant there are 5 places so we have: $5 \times 21^{5}$ words. But that's the number of words when the place of the vowel is fixed so because there are $6$ places multiply the number by $6$ and the total amount would be:
$$30 \times 21^5 \text{ words with only 1 vowel}$$
Words containing 2 vowels
Simularly as the previos case, except now we have 2 places for vowels and 4 for consonants an the totat amount would be: $5^2 \times 21^4$. But the 2 places can be anywhere in the words, so there are $\binom{6}{2} = 15$ places to place them. So the total amount would be:
$$15 \times 5^2 \times 21^4 \text{ words containg exactly 2 vowels}$$
Now just add all those numbers:
$$21^6 + 30 \times 21^5 + 15 \times 5^2 \times 21^4 = 21^4 (21^2 + 21 \times 30 + 15 \times 5^2) = 21^4 (441 + 630 + 375) = 281219526 \text{ words with at most 2 vowels}$$
