CAT Counting: Letters and Words

Question
If all words with 2 distinct vowels and 3 distinct consonants were listed alphabetically, what would be the rank of “ACDEF’?
A.      4716
B.      4720
C.      4718
D.      4717
Correct Answer
Choice C
Explanatory Answer
The first word would be ABCDE. With 2 distinct vowels, 3 distinct consonants, this is the first word we can come up with.
Starting with AB, we can have a number of words
AB __ __ __. The next three slots should have 2 consonants and one vowel. This can be selected in 20C2and 4C1 ways. Then the three distinct letters can be rearranged in 3! Ways.
Or, number of words starting with AB = 20C2* 4C1 * 3! = 190 * 4 * 6 = 4560
Next, we move on to words starting with ACB
ACB __ __. The last two slots have to be filled with one vowel and one consonant. = 19C1 * 4C1. This can be rearranged in 2! Ways.
Or, number of words starting with ACB =  19C1 * 4C1* 2 = 19 * 4 * 2 = 152
Next we move on words starting with ACDB: There are 4 different words on this list – ACDBE, ACDBI, ACDBO, ACDBU
So, far number of words gone = 4560 + 152 + 4 = 4716
Starting with AB
4560
Starting with ACB
152
Starting with ACDB
4
Total words gone
4716
After this we move to words starting with ACDE, the first possible word is ACDEB. After this we have ACDEF.
So, rank of ACDEF = 4718
Answer Choice (C)

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