CMSC 250 Extra Counting Practice ANSWERSEpp Chapter 6
In this set of answers, I have taken the value to an actual integer. You will not need to do this on the
exam for these larger numbers because calculators are not allowed, and I don’t want to spend time testing
if you know how to multiply. I have done it here so that if you used a different method, it should still
evaluate to reach the same integer value.
1. Answer the following about the sequence of numbers between 5 and 125 (inclusive):
a. How many integers are between 5 and 125 (inclusive)?
ANSWER: 125 −5 + 1 = 121
b. How many integers are between 5 and 125 (inclusive) which are divisible by 3?
ANSWER: 2 ∗3,3∗3, ..., 40 ∗3,41 ∗3 = 6,9, ..., 120,123
41 −2 + 2 = 40
c. How many integers are between 5 and 125 (inclusive) which are divisible by 5?
ANSWER: 1 ∗5,2∗5, ..., 24 ∗5,25 ∗5 = 5,10, ..., 120,125
25 −1 + 1 = 25
d. How many integers are between 5 and 125 (inclusive) which are divisible by both 3 and 5?
ANSWER: 1 ∗15,2∗15, ..., 7∗15,8∗15 = 15,30,105,120
8−1 + 1 = 8
e. How many integers are between 5 and 125 (inclusive) which are divisible by either 3 or 5?
ANSWER: n(A∪B) = n(A) + n(B)−n(A∩B) = 40 + 25 −8 = 57
f. How many integers are between 5 and 125 (inclusive) which are divisible by neither 3 nor 5?
ANSWER: Total number - number that are divisible = n(U)−n(A∪B) = 121 −57 = 64
g. How many integers are between 5 and 125 (inclusive) which are divisible by 3 but not divisible
by 5?
ANSWER: number divisible by 3 - number of those that are divisible by 5 = n(A)−n(A∩B) =
40 −8 = 32
2. For the next set of questions, assume you have a of 6 dogs. Some dogs are big (marked with a b),
some are middle sized (marked with a m), and some are small (marked with an s). These dogs are
{Alpi (b),Bingo (s), Congo (b), Delfi(s), Elf (s), Fred (m)}. Answer the following questions about
the dogs in your kennel.
a. How many different ways can you select two dogs to take for a walk at the same time?
ANSWER: 6
2=6!
4!2! = 15
b. How many different ways can you select two dogs to take for a walk at the same time assuming
you can’t handle 2 big dogs at the same time?
ANSWER: Partition and use Addition Rule
Two non-big dogs + One big and one non-big dog
4
2+2
14
1= 6 + 8 = 14
c. How many different ways can you assign the dogs all to leashes (assuming you have 6 leashes in
6 different colors)?
ANSWER: Steps and use multiplication rule
Select the leash for the first dog = 6
∗Select the leash for the second dog = 5
∗...
∗Select the leash for the last dog = 1
= 6!
