MAT 243 Test 3 Practice
1. You have 6 pennies, 3 dimes and 2 quarters in your pocket. You randomly pull out
three coins from your pocket. Find the probability that they add up to at least 50
cents.
2. There are
n
married couples. How many of the 2
n
people must be selected in
order to guarantee that one has selected a married couple?
3. If we flip a coin 8 times, what is the probability that the 5th Head happens on the
last flip?
4. How many people must be in a room to guarantee that there are at least 4 people
who were born in the same month?
5. Find
∑
=
⋅
n
k
k
knC
0
8).(
6. A bag contains 100 apples, 100 bananas, 100 oranges and 100 pears. If someone
randomly picks a fruit out of the bag every second, how long will it be before she
is assured of having at least a dozen pieces of fruit of the same kind? How long
will it be before she is assured of having at least a dozen bananas?
7. Let A be the set of all ternary strings of length 9 that start with a 1, B be the set
of all ternary strings of length 9 that end with a 2, and let C be the set of all
ternary strings of length 9 that has a 0 in the middle position. How many ternary
strings of length 9 are there that either start with a 1, end with a 2 or has a 0 in
the middle position?
8. Prove that in any group of n people, there are at least 2 who have shaken hands
with the same number of people. (Shaking hands with yourself does not count)
9. In a box there are 6 white, 5 blue and 4 red marbles. If we pick a total of 5
marbles without replacement, how many different ways can we pick
(a) exactly 3 white marbles?
(b) at least 2 blue marbles?
(c) exactly 2 white marbles and 1 blue marbles?
10.How many different 10 letter words contains the string NICE if letters can be
repeated?
11.How many 10 digit numbers contains the string 123 if numbers cannot be
repeated?
12.There are 100 people at a party. Each person has an even number (possibly 0) of
acquaintances. Prove that there are three people at the party with the same
number of acquaintances.
13*. Prove that any given 52 integers, there exist two of them whose sum or else
whose difference is divisible by 100.