Find the number of seven lettered words each consisting of three vowels and four consonants which can be formed using the letters of the word $'DIFFERENTIATION'$.

The word DIFFERENTIATION has 7 vowels (4 I's, 2 E's, 1 A, 1 O) and 8 consonants (2 D's, 2 F's, 2 N's, 2 T's, 1 R). To form a 7-letter word with 3 vowels and 4 consonants, we first select the letters: 3 vowels can be chosen from the multiset in 7 ways, and 4 consonants can be chosen from their multiset in 11 ways. The number of arrangements for each selection is 7 factorial divided by the product of the factorials of the repetitions. Multiplying the selections and the arrangements gives 7 times 11 times (7 factorial divided by (2 factorial times 2 factorial)) = 325710.

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