Math Circles: Counting
November 20, 2012
Name: _________________________
Counting Exercises
1.
The Hawaiian alphabet consists of 5 vowels and 7 consonants. How many four letter words are there
with the first and third letter being consonants and the second and fourth being vowels?
2.
There are seven women and four men. How many ways can we form a committee with four people, two
of each gender, such that Mr. and Mrs. Bing cannot both be on the committee at the same time?
3.
Find the number of all 20-digit numbers in which no two consecutive digits are the same.
4.
There are 9 students, three from homeroom A, three from homeroom B, and three from homeroom C.
They bought a block of nine seats for their school’s homecoming game. If three seats are randomly
selected for each class from the nine seats in a row, how many ways can they be arranged in row of nine
seats?
5.
How many ways are there to fill a box with a dozen donuts chosen from five different varieties with the
requirement that at least one donut of each variety is picked?
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Math Circles: Counting
November 20, 2012
Name: _________________________
6.
There are 16 books in the language section of the library. 5 of them are Spanish books, 4 are French
books, and 7 are English books. How many ways can we choose a pair of books that are not both the
same language?
7.
A restaurant offers the following special meal for a fixed price. The customer can choose 1 of 3 salads, 1
of 6 main courses and 1 dessert from the available choices. Suppose that for dessert, the customer can
choose either 1 of 4 sundaes or 1 of 3 different gourmet coffees, and if coffee is chosen, no matter
which one, the customer gets to choose 1 of 3 different types of cookies with it. How many different
meals can a customer create on this special?
8.
How many ways are there to distribute nine balls into six distinct boxes with the first two boxes
collectively having at most four balls if:
(a) the balls are identical?
(b) the balls are distinct?
9.
Two baseball teams A and B play a “best of five” series of games. That is, the winner is the team to first
win 3 games. Find the number of different possible outcomes in which A wins the series.
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Math Circles: Counting
10.
November 20, 2012
Name: _________________________
How many arrangements of the digits 1,1,2,2,3,4,5,5 are there
a) with no restrictions?
b) so that the digits 3 and 5 do not appear in consecutive positions?
11.
How many permutations of the 9 digits 1, 2, 3, 4, 5, 6, 7, 8, 9 have an even number in the first position
and one of the digits 2, 3, or 4 in the last position?
12.
How many ways are there to distribute 20 (identical) sticks of red licorice and 15 (identical) sticks of
black licorice among five children?
13.
Count the number of ways to distribute 12 identical pens, 10 identical pencils, and 9 identical notebooks
among 3 students so that each student has at least 2 pens, 2 pencils and 1 notebook.
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Math Circles: Counting
November 20, 2012
Name: _________________________
14.
If 4 standard (6-sided) fair dice are rolled, then what is the probability that the sum of the numbers
shown is even?
15.
How many ways are there to give a dozen different pens to three people so that one person receives
twice as many pens as the other two combined?
16.
In how many ways can we colour n distinct objects with FOUR colours, such that all the colours are
used?
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Math Circles: Counting
17.
November 20, 2012
Name: _________________________
a) What is the maximum number of pieces of pizza that a person can obtain by making 5 straight cuts
with a pizza knife? (Note: the cuts need not pass through the center of the pizza. The pieces need not be
the same size.)
b) Suppose we generalize this by making n straight cuts. What is the maximum number of pieces of pizza
that a person can obtain by making n straight cuts with a pizza knife?
18.
Using a combinatorial argument, prove that for n>=0,
a) ( )
( )
b) ( )
( )
( )
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