6. Express the trinomial coefficient of the previous problem in six ways as a product of two binomial coefficients.

7. How many combinations are there of 1, 2, 3, or 4 elements from a set of 5 elements?

8. How many non-empty proper subsets are there of a set of n elements? That is, how many combinations are there of 1, 2, ... , or n - 1 elements?

9. Express the coefficient of x³y⁷w² in (x + y + z + w)¹² in six different ways as a product of two binomial coefficients.

10. Express the coefficient of x²y³z⁴w² in (x + y + z + w)¹¹ in six different ways as a product of three binomial coefficients.

11. Find the coefficient of x²y⁹z³w in (2x + y - z + w)¹⁵.

12. Find the coefficient of x^ryzw in (x + y + z + w)^r+3.

13. Show that x⁵y²z⁹ has the same coefficient in (x + y + z + w)¹⁶ as in (x + y + z)¹⁶.

14. What is the relation between the coefficient of xy⁷z² in (x + y - z)¹⁰ and its coefficient in
(x - y + z + w)¹⁰?

15. Let a, b, and n be positive integers, with n > a + b. Show that

16. Express the coefficient of x²y⁴z⁶ in (x + y + z)¹² as the sum of three of the trinomial coefficients in the expansion of (x + y + z)¹¹.

17. What is the sum of all the trinomial coefficients in (x + y + z)¹⁰⁰?

18. What is the sum of the coefficients in each of the following:

(a) (x + y - z)¹⁰⁰?

(b) (x - y + z - w)¹⁰⁰?

19. List the even permutations of 1, 2, 3, 4.
 
 

Tuesday, May 12, 1998