### MAT 336 Stony Brook University Spring 2004

## 15-minute Quiz, April 16, 2004

Total score = 10 points. Show all your work on this sheet.
DO PROBLEM 1 **OR** 2. **CROSS OUT** THE PROBLEM YOU ARE NOT DOING.

- Two people, A and B, are gambling on the flips of a coin. The coin
will be flipped five times in a row. If there are 3 or more Heads, A wins.
If there are 3 or more Tails, B wins. They have each put up $10 to play,
with the understanding that the winner gets $20.

Just after the third toss of the coin, the police
raid the gambling den and the game has to stop. The first three tosses
came up Tails, Heads, Heads. How can the $20 be split fairly between
the two players?

Follow Pascal's reasoning to solve this problem.

- Let the binomial coefficient
`C(n,k)` be (by definition)
the coefficient of `x`^{k} in the expansion of
`(x+1)`^{n}. Prove by induction that
C(n+1,k) = C(n,k-1) + C(n,k).

Explain your argument carefully.