1a. How many arrangements are there of all the letters in INDIVIDUAL?
1b. How many arrangements of the letters in INDIVIDUAL have all three I’s adjacent?
1c. How many arrangements of the letters in INDIVIDUAL have no I’s adjacent?
2. Suppose that you draw five cards from a standard deck of 52.
a. How many ways can you draw exactly 2 hearts?
b. How many ways can you draw at most 2 hearts?
c. How many ways can you draw exactly 2 hearts and 3 clubs?
3. Determine the coefficient of x8y7 in the following expansions:
a. (x + y)15
b. (-3x + y)15
c. (10x - 2y)15
4. Determine the number of integer solutions of x1 + x2 + x3 + x4 = 17, where a. xi ≥ 0, 1 ≤ i ≤ 4
b. x1, x2 ≥ 3 and x3, x4 ≥ 1
c. xi ≥ -2, 1 ≤ i ≤ 4
d. x1, x2, x3 0 and 0 < x4 ≤ 10
5. Prove that if we select 101 integers from the set S = {1, 2, 3, . . . , 200}, there exist m, n in the selection where the greatest common divisor of m and n is 1. Hint: Prove by the Pigeonhole Principle
