2. Find each of the following:

(a) The coefficient of a¹³b⁴ in (a + b)¹⁷.

(b) The coefficient of a¹¹ in (a - 1)¹⁶.

(c) The coefficient of a⁶b⁶ in (a - 3b)¹².

3. Find integers a, b, and c such thatfor all integers n.

4. Find integers p, q, r, and s such thatfor all
integers n.

5. Prove that

6. Given that k is a positive integer, prove that

7. Find

8. Find

9. Prove that

10. Prove that

11. Let m be a positive integer and r a non-negative integer. Expressin terms of a binomial coefficient

12. In the original definition ofas a binomial coefficient, it was clear that it was always an integer. Explain why this is still true in the extended definition.

13. Show thatfor integers a, b, and n, with

14. Given that n = a + b + c + d and that a, b, c, and d are non-negative integers, show that

Monday, June 22, 1998