Programming Team Independent Study Homework Week #1: Combinatorics

Programming Team Independent Study Homework Week #2: Combinatorics

n k n k 1) Evaluate the sum (1)  10 . k 0 k 

2) Use combinatorial reasoning to prove the identity below in the form given:

n n  3 n 1 n  2 n  3               . k   k  k 1  k 1  k 1

3) Prove that for a positive integer n

n n n n1 n    2   3   ...  (1) n   0 1 2 3 n

4) By integrating the binomial expansion, prove that for a positive integer n,

1 n 1 n 1 n 2n1 1 1       ...     2 1 3 2 n 1n n 1

5) Every day a student walks from her home to school which is located 10 blocks east and 14 blocks north from home. She always takes a shortest walk of 24 blocks.

(a) How many walks are possible? (b) Supposed that 4 blocks east and 5 blocks north of her home lives her best friend whom she meets each day on her way to school. How how many different walks are possible? (c) Suppose, in addition, that 3 blocks east and 6 blocks north of her friend's house there is a park where the two girls stop each day to rest and play. How how many different walks are there? (d) Stopping at the park to rest and play the two students often get to school late. To avoid the tempation of the park, our two students decide never to pass the intersection where the park is. How how many different walks are there?

6) Determine the number of permutations of {1,2,3,4,...,8} in which exactly 4 integers are in their natural position. (Feel free to write a program to double check your answer.)

7) At a party 7 men check their hats. In how many ways can their hats be returned so that

(a) no man receives his own hat? (b) at least one of the men receives his own hat? (c) at least two of the men receive their own hats? 8) Use combinatorial reasoning to derive the identity

n n n  n  n n!  Dn   Dn1   Dn2  ...   D1   D0 0 1 2 n 1 n

Note: We define D0 = 1 for the purposes of this identity.

9) Prove the following about the Fibonacci numbers, F(n):

(a) F(n) is even iff n is divisible by 3. (b) F(n) is divisible by 3 iff n is divisible by 4 (c) F(n) is divisible by 4 iff n is divisible by 6 (d) F(n) is divisible by 5 iff n is divisible by 5. (e) Make a conjecture about when F(n) is divisible by 7 and prove it.

10) Solve the recurrence relation:

H(n) = H(n-1) + 9H(n-2) + 9H(n-3) for n > 2 with H(0) = 0, H(1) = 1, H(2) = 2

11) Solve the recurrence relation:

H(n) = 4H(n-1) + 3x2n, for n > 0 H(0) = 1