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?

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?