# Thread: Combinations and Permutations

1. ## Combinations and Permutations

I was having a problem with some questions.

1. A penny is tossed 60 times yielding 45 heads and 15 tails. In how many ways could this have happened so that there were no consecutive tails?

Thanks for your help!

2. Originally Posted by BACONATOR
I was having a problem with some questions.

1. A penny is tossed 60 times yielding 45 heads and 15 tails. In how many ways could this have happened so that there were no consecutive tails?

Thanks for your help!
Let's say the tails occur on tosses x1, x2, ..., x15, listed in ascending order.

The requirement that the tails are not consecutive means that
1 <= x1 < x2-1, x2 < x3-1, x3 < x4-1, ..., x14 < x15-1, x15 <= 60.

An equivalent set of inequalities is
1 <= x1 < x2-1 < x3-2 < x4-3 < ... < x14-13 < x15-14 <= 46,

so selecting the positions of the tails is equivalent to selecting the numbers x1, x2-1, x3-2, ..., x15-14
(but now without any restriction on adjacency),

which can be done in C(46,15) ways.