04 Advanced Problem Solving Combinatorics - IMSA

3 downloads 261 Views 101KB Size Report
1)a) Find the number of combinations (order doesn't matter) of ten letters using {A , B, C} that contain at least one A, two B's, and 3 C's. b) Find the number of ...
Advanced Problem Solving Combinatorics Fall 2013

Name: ________________________

1)a) Find the number of combinations (order doesn’t matter) of ten letters using {A, B, C} that contain at least one A, two B’s, and 3 C’s. b) Find the number of strings (order does matter) with the same requirements.

2) An election takes place between two candidates, A and B. Assume A receives a votes, and B receives b votes, with a > b. Assume the votes are counted one at a time (in absolutely random order). What is the probability that during the count A never trails? What is the probability that A is always ahead (never trails, never tied)? (Answer in terms of a and b.)

3) Use properties of binomial coefficients to determine the exact value (in terms of n) of ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ 1 ⎛ n⎞ 1 ⎛ n⎞ 1 ⎛ n⎞ a) ⎜ ⎟ + 2 ⎜ ⎟ + 3 ⎜ ⎟ +!+ n ⎜ ⎟ b) ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ +!+ n + 1 ⎜⎝ n ⎟⎠ ⎝ 1⎠ ⎝ 2⎠ ⎝ 3⎠ ⎝ n⎠ ⎝ 0⎠ 2 ⎝ 1⎠ 3 ⎝ 2⎠

4) Find the number of partitions of 12 (into any number of positive parts).

5)a) Explain why the number of partitions of k into n nonnegative parts is the same as the number of partitions of n + k into n positive parts. b) Explain why the number of partitions of k into at most n positive parts is the same as the number of partitions of k into exactly n nonnegative parts. (Note: these two together prove that P(k, 1) + P(k, 2) + ! + P(k, n) = P(n + k, n).

6) Explain why the number of partitions of k into at most four parts is the same as the number of partitions of k into parts of size at most four. (Hint: stare at the diagram to the right, which basically proves the statement if four is replaced by three.)