1. Find the coefficient of x^24 in (x^3 + . . . .+ x^6)^5.

2. Consider the problem of counting the ways to select 25 objects from 8 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 MATHEMATICAL are there in which ME appear together but the ME is not immediately followed by an A (no MEA)

4. What is the probability that a 9-card subset out of the 52-card deck has two three-of-a-kinds but no four-of-a-kind and no pairs?

5. How many 6-digit decimal sequences are there with exactly r 3'S ( for some r, 1 <= r <= 5)?

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

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

(ii) begins or ends with an I, and

(iii) the consonants are in alphabetical order.

7. You have 6 friends. How many ways are there to invite a different subset of two of these friends over for dinner on 4 successive nights?

1. Find the coefficient of x^30 in (x^2 + . . . .+ x^8)^7.

2. Consider the problem of counting the ways to distribute 29 identical objects in 7 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 sequences of length 6 are formed from the 26 letters without repetition where the first or last letter (possibly both) must not be vowels (a,e,i,o,u)?

4. How many 10-card hands are there chosen from a standard 52-card deck in which there are exactly two 4-of-a-kinds; no pairs or 3-of-a-kinds?

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 the letters in DISAPPOINTING have ALL of the following properties:

(i) the I's are non-consecutive;

(ii) the arrangements starts with a G, and

iii) the consonants are not in alphabetical order

7. Suppose a coin is tossed 14 times and there are 3 heads and 11 tails. How many such sequences are there in which there are at least 6 tails in a row?

1. Find the coefficient of x^23 in (x+ x^2+ x^3+. . . x^9)^5.

2. Consider the problem of counting the ways to distribute 31 votes among 6 candidates with at least two votes for each candidates.

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. What is the probability that a 7-card hand (from a 52-card deck) has four of some kind and the other three cards are each of a different kind?

4. How many arrangements of the 26 different letters (with repeats allowed) are there which contain exactly two A's and exactly one N?

5. How many arrangements of PREPARING are there in which each P is followed by a vowel(A,E,I)?

6. How many arrangements of letters in ARITHMETIC have ALL of the following pr operties:

(i) the vowels are non-consecutive,

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

(iii) begins with a consonant?

7. How many sequences of 5 A's, 6 B's, and 5 C's are there in which the first A comes somewhere before the first B?