# Combination of letters in alphabet

An alphabet has 7 vowels and 15 consonants. How many ways are there to order the letters of the alphabet so that each vowel is followed immediately by at least 1 consonants?

I was thinking that pairing the vowels with a consonant and then finding the ways to arrange 7 pairs and 8 single letters but I'm not sure how to do that. My try was $P(22,15)$ or $C(22,15)$, but neither worked. I thought about using the dots and dividers method, but that doesn't take order into count, so I don't think that will be right.

Any help would be awesome, thank you!

We can arrange the $15$ consonants in $15!$ ways. To ensure that each vowel is followed immediately by at least one consonant, we insert the vowels in seven of the fifteen spaces to the immediate left of a consonant, which can be done in $\binom{15}{7}$ ways. Finally, we can arrange the seven vowels in the chosen spaces in $7!$ ways. Hence, the number of permissible arrangements is $$15!\binom{15}{7}7!$$