1. Find the coefficient of x^32 in (x^2+ . . . + x^6)^8.

2. Consider the problem of counting the ways to distribute 31 identical objects in 5 boxes with at least three objects in each box.

a) Model this problem as an integer-solution-of-equation problem.

b) Model this problem as a certain coefficient of a generating function.

c) Solve this problem.

3. How many 4-digit campus telephone numbers (4-digit decimal sequences) are there in which the digit 6 appears at most twice (maybe not at all)?

4. Find the probability that a 9-card hand (from a 52-card deck) has exactly 2 three-of-a-kinds (no 4-of-a-kinds and no pairs).

5. How many arrangements 8 letters long are there formed from A’s, B’s and C’s such that each letter appears at least twice (you must break into cases).

6. How many arrangements of letters in SUBSTITUTING have ALL of the following properties:

(i) there are at least 3 letters inbetween each T,

(ii) vowels are not alphabetical order, and

(iii) the arrangement starts with a consonant.

7. How many ways are there to pick an (unordered) subset of 6 cards from a standard 52-card deck so that the subset contains at least one Ace, at least one King, at least Queen, and at least one Jack?

1. Find the coefficient of x^28 in (x^3+ . . . + x^8)^5.

2. Consider the problem of counting the ways to distribute 31 identical objects in 6 boxes with at least three objects in each box.

a) Model this problem as an integer-solution-of-equation problem.

b) Model this problem as a certain coefficient of a generating function.

c) Solve this problem.

3. How many arrangements of INCONSISTENT are there in which NE appear consecutively or NO appear consecutively but not both NE and NO are consecutive?

4. How many 5-letter sequences (formed from the 26 letters in the alphabet, with repetition allowed) contain exactly two A's and exactly one N?

5. Find the probability that an 11-card hand (from a 52-card deck) has exactly 4 pairs (no 4-of-a-kinds and no 3-of-a-kinds).

6. How many arrangements of letters in DISAPPEARANCES have ALL of the following properties:

(i) there are at least two letters between each A,

(ii) begins with a consonant, and

(iii) the consonants are in alphabetical order.

7. How many integer solutions are there to 2x1 + 2x2 + 2x3 + x4 + x5 = 9 with xi>= 0? (Hint: break into cases).

1. Find the coefficient of x^37 in (x^4 + . . . .+ x^7)^6.

2. Consider the problem of counting the ways to select 15 objects from 4 types with at least two of each type.

a) Model this problem as an integer-solution-of-equation problem.

b) Model this problem as a certain coefficient of a generating function.

c) Solve this problem.

3. How many arrangements of length 12 formed by different letters (no repetition) chosen from the 26-letter alphabet are there that contain the five vowels (a,e,i,o,u)?

4. Find the probability that a 12-card subset out of the 52-card deck has three three-of-a-kinds but no four-of-a-kind and no pairs?

5. How many ways are there to arrange 12 (distinct) people in a row so that Dr. Tucker is 3 positions away from Dr. Stanley (i.e., 2 people are inbetween Dr. Tucker and Dr. Stanley), e.g., . . . . T _ _ S . . . .?

6. How many arrangements of letters in INCONSISTENT have ALL of the following properties:

(i) the vowels are non-consecutive,

(ii) the consonants are not in alphabetical order, and

(iii) the arrangement ends with a vowel.

7. How many subsets of four numbers from the set 2,3,4,5,6,7,8,9,10,11, 12,13,14 are there in which the sum of the largest and smallest number in the subset is 15?