AMS 301 SECOND TEST Test A Fall, 2009
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.(BR> 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 r 3's.
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.

AMS 301 SECOND TEST Test A Fall, 2008
1. Find the coefficient of x^29 in (x^2+ x^3+ x^4+ x^5+ x^6)^8.
2. Consider the problem of counting the ways to distribute 23 identical objects into 6 different boxes with at least three objects for each box.
a) Model this problem as an integer-solution-of-equation-problem.(BR> b) Model this problem as a certain coefficient of a generating function.
c) Solve this problem.
3. How many 6-digit decimal (0,1,2, . . . 9) sequences are there that start with at least one 3 and end with a 3?
4. Find the probability that 8-card hand (from a 52-card deck) has one three-of-a-kind (no 4-of-a-kinds and no pairs).
5. How many ways are there to arrange n (distinct) people in a row (where n >= 8) so that Dr. Tucker is 4 positions away from Dr. Kenny (i.e., 3 people are inbetween Dr. Tucker and Dr. Kenny)?
6. How many arrangements of letters in SUBSTITUTING have ALL of the following properties:
(i) S's non consecutive, (ii) vowels not in alphabetical order, and (iii) the 3 T's are consecutive.
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?
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.