The number of ways of choosing m coupons out of an unlimited number of coupons bearing the letters A, B and C so that they cannot be used to spell the word BAC, is

The word BAC cannot be spelt if the m selected coupons do not contain atleast one of A, B and C.Number of ways of selecting m coupons which are A or B = 2m.This also includes the case when all the m coupons are A or all are B.Number of ways of selecting m coupons which are B or C = 2m.This also includes the case when all the m coupons are B or all are C.Number of ways of selecting m coupons which are C or A = 2m.This also includes the case when all the m coupons are C or all are A.Number of ways of selecting m coupons when all are A = 1m.Number of ways of selecting m coupons when all are B = 1m.Number of ways of selecting m coupons when all are C = 1m.∴ Required number = 2m + 2m + 2m – (1m + 1m + 1m)                                = 3 ⋅ 2m – 3 ⋅ 1m = 3 (2m – 1).