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 ^{20}C_{2} and ^{4}C_{1} ways. Then the three distinct letters can be rearranged in 3! Ways.
Or, number of words starting with AB = ^{20}C_{2} * ^{4}C_{1} * 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. = ^{19}C_{1} * ^{4}C_{1}. This can be rearranged in 2! Ways.
Or, number of words starting with ACB = ^{19}C_{1} * ^{4}C_{1} * 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
Correct Answer : 4718
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