2015/16 AHSMC Part 1 Problem 6

Problem 6 of 16

Ellie wishes to choose three of the seven days (Monday, Tuesday, …, Sunday) on which to wash her hair every week, so that she will never wash her hair on consecutive days. The number of ways she can choose these three days is:

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Solution

Ellie can choose one of the following triplets: $(M, W, F)$ , $(M, W, Sa)$ , $(M, R, Sa)$ , $(T, R, Sa)$ , $(T, R, Su)$ , $(T, F, Su)$ , $(W, F, Su)$ . There are seven possibilities.

Here is an alternate solution: In any such choice of three wash days, exactly one of them must be followed by two non-wash days. The choice of this day will determine the other two wash days. There are seven possibilities and thus the answer is (B).

University of Alberta

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