Question

How many four letter code words can be formed from the letters in the word "MIRAGE" if no letter is repeated, and the second-to-last letter must be a vowel?​

Answer 1

180

Explanation:

6 different letters.

we need to pick groups of 4.

no repetitions, but the sequence matters (code words), as e.g. RAGE is different to GEAR, although they contain the same letters.

so, we need basic permutations (instead of combinations) :

P(6, 4) = 6! / (6 - 4)! = 6! / 2! = 6×5×4×3 = 360

that is simply because regularly we would have 6 choices for the first letter, then 5 for the second, 4 for the third, and 2 for the fourth letter.

the second to the last letter is the second letter from the left in a 4-letter word.

so, we have 3 vowels for that second position.

normally, such a restriction would mean

6×3×4×3 = 216 possibilities.

but we have to distinguish the 2 cases that we pick a vowel for the first position - or not.

if not, we have 3 consonants for the first, 3 vowels for the second position as options.

if yes, we have 3 vowels for the first and 2 vowels for the second position.

that means we get

3×3×4×3 + 3×2×4×3 = 108 + 72 = 180

possibilities.

this makes also sense, when we simply say that this restriction eliminates half of our possible permutations (all with a consonant in the second position) : 360/2 = 180.