Test 1 – Review
Covering Sections: 2.1-2.7, 3.1-3.2, 5.1-5.4
2.1-2.7: Permutations, Combinations of sets and multi-sets and finite probability.
Good problems include 1-64 in Chapter 2 – these are the basics of what we need the rest
of the semester. Especially # 11, 14, 26, 32-35, 37, 38, 51, 52, 55, 60, 63 & 64.
3.1-3.2 Pigeonhole Principle and the Strong Pigeonhole Principle
Good problems include 1-19 in Chapter 3 – especially # 4, 7, 8, 11, 12, 15 – with 26 – 28 a
bit more challenging than expected – but fun (??).
5.1-5.4: Pascal’s Triangle, Binomial Theorem and the Multinomial Theorem.
Good problems include # 11, 13, 15, 25, & 27-29.
Other problems:
Combinations, Permutations:
How many six digit numbers
a. consist of six different digits?
b. consist of five different digits?
c. consist of three odd and three even digits?
d. have six different digits with no two even digits adjacent?
e. have four distinct odd digits and two distinct even digits which are not
adjacent?
A box contains 2 pennies, 4 nickels, and 6 dimes. Six coins are drawn without
replacement, with each of the 12 coins having the same probability of being chosen.
What is the probability that the value of the coins is at least 50 cents?
Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
a. How many five element subsets does the set have?
b. How many subsets of S have an odd number of members?
c. How many subsets of S have 1 as a member?
d. How many subsets have 1 as a member and do not have 2 as a member?
A city council is composed of 5 liberals and 4 conservatives. A delegation of three is to
be selected to attend a convention.
How many delegations are possible?
What is the probability that the delegation is all liberals?
What is the probability that the delegation has a liberal tilt – that is 2 liberals and
one conservative?
Pigeonhole Principle:
-
Cells of a 15×15 square grid have been painted in red, blue and green. Prove that there
are at least two rows of cells with the same number of squares of at least one of the
colors.
-
If 9 people are seated in a row of 12 chairs, then some consecutive set of 3 chairs are
filled with people.
-
Given 12 distinct 2-digit integers. Prove there are some two whose difference - a 2-digit
number - has equal digits.
-
Given any 10 4-element subsets of an 11-set, some two of the subsets intersect in at least
two elements.
-
My wife and I recently attended a party at which there were four other married couples.
Various handshakes took place. No one shook hands with oneself, nor with one's
spouse, and no one shook hands with the same person more than once. After all the
handshakes were over, I asked each person, including my wife, how many hands he (or
she) had shaken. To my surprise each gave a different answer. How many hands did
my wife shake?
Binomial Theorem:
Prove that for all positive integers n
By considering the expansion of (1+x)n and (1-x)n show: