In an experiment a fair coin is tossed 7 times and the face

In an experiment, a fair coin is tossed 7 times and the face that appears (H for head or T for tail) for each toss is recorded. How many elements of the sample space will start and end with a head and include a total of exactly three heads?

Solution

Each \"element of the sample space\" is a sequence of 7 results, each result being T or H.

The 3 heads can\'t be the starting or ending pair, or the sequence would both start and end with a pair, because all the rest must be tails.

But both the heads can\'t be in the middle seven positions, or again the sequence would start and end with a pair (both pairs being tails in this case).

So we have a couple of cases.

Case 1: One H in one of the middle 7 positions (7 possibilities), which requires the other H to be in any of the remaining 4 positions (4 possibilities). Case 1 gives us
7 * 4 = 28 possible sequences.

Case 2: No Hs in the middle 7 positions. But then they must be on opposite ends, where each has 2 possible positions (at the end or adjacent to the end). So case 2 gives us
2 * 2 = 4 possible sequences.

A total of
28 + 4 = 32 sequences satisfy these criteria.