1) The board of directors of a pharmaceutical corporation has 10 members. An upcoming stockholder's meeting is scheduled to approve a new slate of company officers (chosen from the 10 board members). A) 4 (Presendent, Vice Presendent, secretary, and treasurer) positions needs filled. How many possible ways are there to do it? 10 x 9 x 8 x 7 B) Three members of the board of directors are physicians. How many slates from part (A) have a physician nominated for presendency? 3 x 9 x 8 x 7 ???? C) How many slates have exactly one physician? 3 x 7 x 6 x 5 D) How many slates have at least one physician? 3 x 9 x 8 x 7 2) There are 3 cities: a, b, c. City a has two roads that go to C and 4 roads that go to b. City b has 3 roads that go to c. A) How many ways can you get from a to c? 2 + (4x3) B) How many round trips from a to c are there such that the return trip is at least partially different than the route taken to get there? (2 + (4x3)) x (2 + (4x3)) - (2 + (4x3)) ???? 3) If n is a positive integer and n is greater than 1, prove that c(n, 2) + c(n-1/2) is a perfect square. I have no idea 4) A gym coach must select 11 seniors to play on a football team. If he can make his selection in 12,376 ways, how many seniors are eligible to play? n! / (11! x (n-11)!) = 12,376 n! / (n-11)! = 12,376(11!) That's all I got 5) How many ways can 10 identical dimes be distributed among five children if: A) There are no restrictions? c(14,10) B) Each child gets at least one dime? c(9,5) C) The oldest child gets at least two dimes? c(12,8) I know those are the answers, but I don't know why. 6) Determine the number of integer solutions of: x1 + x2 + x3 + x4 = 32 where: A) xi >= 0, 1<= i <=4 c(35, 32) B) xi > 0, 1<= i <=4 c(31, 28) Why? C) x1, x2 >= 5, x3, x4 >= 7 c(11, 8) Why?