Math 210 Weekly Proof #8 Chapter 11 - Counting Remember that when writing proofs, you will be graded both on the correctness of your logic and on the clarity of your writing. Use complete sentences, and refer to section 2.3 for general writing advice. n! Recall that Is the number of r-subsets of a set of size n. 1. (4 points) In a particular chess variant, the pieces are placed randomly in a row behind the pawns.

Determine the number of possible ways a player's pieces can follow all of these rules:

There are 8 pieces (a King, a Queen, two Rooks, two Knights, and two Bishops)

The two Bishops must go on opposite colors squares (Note: squares alternate between dark and light)

The King must go between the two Rooks (though the other pieces can also be between them)

The Queen and Knights have no restrictions

2. (6 points) Consider the set {1,-1,2, -2,3,-3,..., n, -n}. Use a combinatorial proof to prove that: )(-x) = (27) k=0 Note: To get full credit you must use a combinatorial proof (by counting a set twice). Do not directly prove the claim by using identities or by evaluating the binomial coefficients.