1. Find the coefficient of x^23 in (x+ x^2+ x^3+. . . x^97^5.

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?

1. Find the coefficient of x^23 in (x+ x^2+ x^3+. . . x^9)^7.

2. Consider the problem of counting the ways to distribute 27 identical objects into 6 boxes with at least 4 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 the 26 different letters (no repeats) are there in which all the consonants appearsbefore all the vowels (a,e,i,o,u)?

4. How many 8-card hands (from a 52-card deck) have exactly 3 pairs and no 3-of-a-kinds and no 4-of-a-kinds.

5. How many arrangements of PEPPERMILL are there in which MP appear consecutively or LP appear consecutvely but not both MP and LP are consecutive?

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

(i) the vowels are non-consecutive,

(ii) the consonants are in alphabetical order, and

(iii) the arrangement ends with a vowel.

7. How many ways are there to distribute 8 different toys among six different children if at most 3 toys are given to the first two children combined?

1. Find the coefficient of x^24 in (x^3+ x^4+ x^5+ x^6)^5.

2. Consider the problem of counting the ways to select 25 objects from 6 types with at least two objects 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. Find the probability that 4-card hand (from a 52-card deck) has no pair each card of a different kind).

4. How many 6-digit decimal sequences are there with exactly

5. How many arrangements of MATHEMATICAL are there in which ME appear together but ME is not immediately followed by an A (no MEA)?

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

(i) at least two letters between each I, (ii) begins or ends (not both) 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 5 successive nights.