AMS 301 SECOND TEST Test A Fall, 2007
1. Find the coefficient of x^23 in (x^2+ x^3+ x^4+ x^5+ x^6+x^7)^5.
2. Consider the problem of counting the ways to distribute 31 votes among 6 candidates with at least two votes for each candidate.
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 6-card hand contains four of some kind (and the other two cards do not form a pair)?
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. How many arrangements of PREPARING are there in which each P is followed by a vowel (A,E,I)?
6. How many arrangements of the letters in ARITHMETIC have ALL of the following 3 properties:
(i) begins with a consonant
(ii) at least 1 consonant between each vowel, and
(iii) the consonants are NOT in alphabetical order
7. How many sequences of 5 A's, 6 B's, and 5 C's are there in which the first A precedes the first B.

AMS 301 SECOND TEST fall,2006
1. Find the coefficient of x^25 in (x + x^2 + . . x^9)^7.
2. Consider the problem of counting the ways to distribute 29 identical objects into 6 boxes with at least 4 objects in each box
a) Model this problem as an integer-coefficient-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 letters of the alphabet (with no repeats) are there in which all the vowels (a,e,i,o,u) appear before all the consonants?
4. How many 7-card hards chosen from the 52 cards in a deck are there containing exactly 3 pairs (no 3-of-a-kind or 4-of-a-kind)?
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 INCONSISTENT have ALL of the following properties:
(i) the vowels are non-consecutive;
(ii) the consonants are in alphabetical order; and
(iii) the arrangement ends with a vowel.
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 5 tails in a row? Hint: Think of such a sequence as a bunch of tails (maybe none), a first head, then another bunch of tails, then a second head, etc.

AMS 301 SECOND TEST fall,2005
1. Find the coef. of x^25 in (x^3 +x^4 + x^5 + x^6)^5.
2. Consider the problem of counting the ways to select 25 objects from 8 types with at least 2 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. What is the probability that a 4-card subset out of the 52-card deck has no pairs (each card has a different value)?
4. How many 6-digit decimal sequences are there with exactly r 3's.
5. How many arranagements of MATHEMATICAL are there in which ME appear consecutively, but they are not immediately followed by A (no 'MEA')?
6. How many arrangements of the letters in STATISTICAL have ALL of the following properties:
(i) at least two consonants between successive vowels:
(ii) the arrangement starts with a vowel; and
(iii) the consonants are in alphabetical order.
7. How many integer solutions are there to 2x1 + 2x2 + 2x3 + x4 + x5 + x6 + x7 = 9?