a) Show that
A counting proof could be fun(?). But any old proof will do.
(Note that the coefficients (1,2,1) in the above are just the elements of the second row of Pascal's triangle. In general, if you take any row of Pascal's triangle and apply all of the coefficients to adjacent entries of a later row in the table, you will get another entry in Pascal's triangle. You don't have to prove this).
b) Not connected to part a) above (I don't think). Consider the two player Problem of Points set up, where the game consists of n rounds, and where player A has won a rounds and Player B has won b rounds (a, b < n)whentheyareforcedtoquit.Let r =2n - 1 - (a + b). Show that according to the Pascal-Fermat solution, the ratio of A's share of the pot to B's share of the pot should be:
That is, all you need is the r'th row of Pascal's Triangle to get the split of the pot, as pointed out by Pascal.