Permutations of "SOCIOLOGICAS" with the vowels in order?

Question

How many permutations of "SOCIOLOGICAS" have the vowels in order? For example: AIIOOOSCLGCS or AISCILOGOCOS. Thanks!

Answer 1

As @DanielBuck mentioned, you should give some context , what you have done , etc.But I’m going to assume you have basic knowledge of permutation and combination. First , read this : In how many ways can we put $k$ sticks between $n$ circles? So let’s assume that the vowels are the circles and the other letters are the sticks. There are $12!$ permutations in total but we have to divide that by $6!$ because the vowels should be in order and there’s only one permutation that the vowels are in order. (Please note that we didn’t count any extra permutations for the non-vowel letters because they didn’t have to be in any order.) So the answer would be $12!/6!$.

Answer 2

SOCIOLOGICAS
First separate the vowels and consonants.
AIIOOO CCGLSS
We have 6 vowels and 6 consonants.
The vowels can go in any of C(12, 6) positions = $$\frac{12!}{(6! * 6!)} = 924.$$
but have to be in order so there is only one way to do that once the positions are chosen.

The consonants go in the other positions,
and 6 unique consonants could be arranged in 6! ways,
but there are 2 C's and 2 S's, so divide that by 2! 2!

The answer is then $$924 * \frac{6!}{(2! * 2!)} = 166320$$

This is also just $$\frac{12!}{6! 2! 2!}$$ once all the 6!'s are canceled.