If all words with 2 distinct vowels and 3 distinct consonants were listed alphabetically, what would be the rank of “ACDEF’?
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 20C2 and 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
The question is "what would be the rank of “ACDEF’?"