In an experiment a fair coin is tossed 11 times and the face

In an experiment, a fair coin is tossed 11 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 with a pair (TT or HH ) or end with a pair (but not both) and have a total of exactly two tails?

Solution

According to the description, it requires that:

It should start with HH, and have 2 tails for the next 9, but the end should not be TT or HH.

That is:

It should start with HH, and have 2 tails for the next 9, but should end with TH or HT.

Thus,

1. There is only one way to do the first 2 items.
2. There are 2 ways to arrange the last 2 items.
3. As one T is used for the last item, the middle 7 items must have 6 H and 1 T. Thus, there are 7!/(6!1!) = 7 ways to do it.

Thus, there are 2*2*7 = 28 such elements. [ANSWER]