1. Find the coefficient of x^23 in (x^3+x^4+x^5+x^6)^4.

2. Consider the problem of counting the ways to select 27 donuts from 5 types with at least three 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 INSTRUCTIVE in which each T is followed by a vowel?

4. How many sequences of length 6 formed from the 26 letters without repetition are there where the vowels (a,e,i,o,u) may only appear in the first or/and last positions (possibly neither)?

5. Find the probability that a 13-card hand (from a 52-card deck) has exactly 3 three-of-a-kinds (no 4-of-a-kinds and no pairs).

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

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

(ii) ends with an A, and

(iii) the consonants are not in alphabetical order.

7. How many ways are there to select 8 donuts from 6 types of donuts if at most 3 donuts are chosen from the first two types combined;

that is, at least 5 donuts chosen from the other four types. (Hint: break into cases)

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 vowel.

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).