Download Math325 practice Test 1 1. Consider 4

Math325 practice Test 1
(on 2.1 to 2.6, 3.1, 3.2, 5.1, 5.2, 5.4, 5.5)
1. Consider 4-digit numbers whose digits are either 1, 2, 3, 4, or 5.
a. How many numbers are there?
b. How many numbers are there if the digits are distinct?
c. How many odd numbers are there if the digits are distinct?
2. In how many ways can seven men and seven women be seated at a round
table if the men and women are to sit in alternate seats?
3. A woman invites a nonempty subset of twelve friends to a party.
a. In how many ways can she do it, if two of the friends are married to each
other and must be invited together or not at all.
b. Repeat (a) if instead the two friends are recently divorced and cannot
both be invited at the same time.
4. You are dealt a hand of poker, that is, a set of 5 cards from the standard deck
of 52 (the order of the cards within a hand does not matter). Assume that all
hands are equally likely. Which is larger, the probability of getting a full
house or the probability of getting four-of-a-kind? What is the ratio of the
probabilities? (Remember: The deck contains 4 cards in each of 13 ranks. A
full house is a hand that has three cards of one rank and two cards of a
different rank. A four-of-a-kind is a hand that has four cards of one rank and
one card of a different rank.)
5. Twelve new students arrive at a certain wizarding academy, and must be
sorted into four different houses, with three going to each house. In how
many ways can this be done? Express your answer in terms of factorials, and
cancel all factorials that occur in both the numerator and the denominator.
6. Twelve students taking a potions class are to be divided into four workinggroups of three students apiece. All we care about are which students are
working together and which students are not. In how many ways can the
students be divided into the groups? Be sure to relate your answer to the
answer to the preceding problem #5.
7. The number of linear permutations of the set {1, 2, 3, 4, 5, 6} is equal to 3!
times 3! times the number of linear permutations of the multiset {o, o, o, e,
e, e}, since we can replace the o’s by the three odd numbers in 3! different
ways and the e’s by the three even numbers in 3! different ways. Verify this
numerically by computing the number of linear permutations of the set {1, 2,
3, 4, 5, 6} and the number of linear permutations of the multiset {o, o, o, e,
e, e}. (You need not list them all; just count them using formulas from the
chapter.)
8. A doughnut store sells six different kinds of donuts. If a box of doughnuts
contains 14, how many different options are there for a box of doughnuts if
it contains at least two powdered doughnuts?
9. Use the pigeonhole principle to show that if you place 33 pebbles on a
chessboard, there must be two pebbles that occupy the same square or are in
adjacent squares (where ‘adjacent’ means ‘horizontally or vertically
adjacent’).
n
10. Evaluate
 (1)
k 0
k
n
2 k 3 n k  
k 
11.What is the coefficient of u 3 v 4 t in the expansion of (u  v  2t ) 8 ?
12.Use Newton’s binomial theorem to approximate 101 / 3