Find the number of 7 lettered words each consisting of 3 vowels and 4 consonants which can be formed using the letters of the word “DIFFERENTIATION”.

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533770

531770

552770

532770

answer is D.

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Detailed Solution

If we assume that each vowel & consonant can only occur only one time, some vowels and consonants appear multiple times.
1. Three same vowel (III only): 1
2. Two same vowels (either II or EE) and one unique vowel: 2C1×3C1×3C2 = 18 3. Three unique vowels: 3! × 4C3=24
∴ There are total of 1+18+24=43 possible combinations.
1. Two times same consonants: 3C2×4C2=18 2. Two same consonants and two unique consonants. 2! × 3C1×4C2×4C2 = 216
3. Four unique consonants. 4! × 5C4 = 120
∴Total possibilities = 18+216+120 = 354
Now, selecting three positions to place the vowels, the total number of words = 43×354×7C3 = 532770
Hence, Option (4) is the correct answer.

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