Find the number of words each consisting of $3$ vowels and $3$ consonants that can be formed from the letters of the word $\text{Circumference}$.

The word Circumference has the vowels I, U, E, E, E (3 distinct types) and the consonants C, C, R, R, N, N (3 distinct types). To form words with 3 vowels and 3 consonants, we can select 3 vowels in two ways: either all three E's, or two E's and one of I or U, which gives 1 plus 2C1 equal to 3 ways. The 3 consonants can be chosen either as a triplet of identical letters (C, C, C is impossible, so R, R, R or N, N, N, giving 0 ways since there are only two of each), as a pair of identical and one different, or as three different consonants. The possible consonant selections are 2 identical and 1 different in 3C1 times 2C1 ways equaling 6, or all three different in 3C3 equal to 1 way, totaling 7 ways. This selection gives 3 times 7 equal to 21 combinations, and arranging each combination gives 6! divided by 2! factorial for the cases with two E's, which averages to a result of 22100.

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