Question

If the number of ways of selecting k coupons one by one out of an unlimited number of coupons bearing the letters a, t, m so that they cannot be used to spell the word mat is 93, then find k.

Answer 1

In order to misspell the word MAT, your k coupons must be missing at least one character. So you either have one letter repeated k times, or you have 2 letters only.

Number of ways to have a letter repeated k times is  3C1=3.

Number of ways in which any two letters are chosen is $3C2(2^{k}-2)$.

Adding the two possible scenarios gives:

$3+3C2(2^{k}-2)=93\\=3+3(2^{k}-2)=93$

$3(2^{k}-2)=90\\2^{k}-2=30\\2^k=32\\k=5$