First slide

Combinations

Question

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

Moderate

A

3 (2^m – 1)
B

3 (2^{m – 1} – 1)
C

3 (2^m + 1)
D

None of these

Solution

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 = 2^m.
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 = 2^m.
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 = 2^m.
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 = 1^m.
Number of ways of selecting m coupons when all are B = 1^m.
Number of ways of selecting m coupons when all are C = 1^m.
∴ Required number = 2^m + 2^m + 2^m – (1^m + 1^m + 1^m)
= 3 ⋅ 2^m – 3 ⋅ 1^m = 3 (2^m – 1).

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