This document provides examples of counting techniques problems involving permutations, combinations, and arrangements. It includes 17 questions on line permutations, circular permutations, pair permutations, and combinations. The questions cover a range of scenarios involving selecting committees, arranging objects, seating arrangements, and more.

1. Counting Techniques
Line Permutation
Q1. A committee of 3 members is to be formed consisting of one representative each from labor, management,
and the public. If there are 2 possible representatives from labor, 2 from management, and 4 from the
public, determine how many different committees can be formed.
Q2. In how many ways can 5 differently colored balls can be arranged in a row?
Q3. In how many ways can 10 people be seated on a bench if only 4 seats are available?
Q4. How many 4-digit numbers can be formed with the 10 digits 0, 1, 2, …, 9 if
A. repetitions are allowed
B. repetitions are not allowed
C. the last digit must be zero and repetitions are not allowed
Q5. Five different science books, six different history books, and two different statistics books are to be
arranged on a shelf. How many different arrangements are possible if
A. The books in each particular subject must all stand together,
B. Only the science books must stand together?
Q6. The letters, "A, B, C, D, E, F, and G" are used to form a five letter secret code. Repetitions are not
allowed. How many different codes are possible?
Q7. The lockers in the gymnasium have 4-digit combination locks. The first digit must be odd, the second
digit must be even, and the last 2 digits can be any number from 0 through 9. How many different lock
combinations exist?
Q8. The president of a country and 4 other dignitaries are scheduled to sit in a row on the 5 chairs represented
above. If the president must sit in the center chair, how many different seating arrangements are possible
for the 5 people?
Q9. In how many arrangements can a teacher seat 3 girls and 3 boys in a row of 6 if the boys are to have the
first, third, and fifth seats?
Q10. If a customer makes exactly 1 selection from each of the 5 categories listed below, what is the greatest
number of different ice cream sundaes that a customer can create?
12 ice cream flavours
10 kinds of candy
8 liquid toppings
5 kinds of nuts
With or without whip cream.
Q11. In how many ways can the letters of the word SPECIAL be arranged using:
A. All the letters?
B. Only 4 letters at a time?
Q12. How many 4-digit arrangements can be created using the digits 5, 7, 8 and 2 if
A. the numbers can be repeated
B. the five must be the first number
C. the five must be the first number and the two the second
Q13. A five alpha-numeric case-insensitive password is chosen so that the first character is not a digit. How
many such passwords are possible?
Q14. How many different ways can an eight-question true-false test are answered?

2. Q15. How many different ways can an eight-question multiple-choice test be answered if each question has five
possible answers?
Q16. How many different three-digit numbers can be formed using the digits 5,6,7,8 and 9 without repetition?
Circular Permutation
Q1. In how many ways can 7 people be seated at a round table if
A. they can sit anywhere,
B. two particular people must sit next to each other?
C. two particular people must not sit next to each other?
Q2. A married couple invites 3 other couples to an anniversary dinner. In how many different ways can they
seated around a circular table couple vise?
Pair Permutation
Q1. Five red balls, two white balls, and three blue balls are arranged in a row. How many different
arrangements are possible, knowing that all the balls of the same color are not distinguishable from each
other?
Q2. How many permutations are there for the letters in the word, STATISTICS?
Q3. Consider a bag of billiard balls containing three identical red balls, a blue ball, and a yellow ball. Select
each of the balls one at a time and arrange them in the order selected. How many different ways can this
occur?
Combination
Q1. In how many ways can a Statistics teacher select five students to work problems on the board from a class
of twenty students?
Q2. A store manager has a choice of two employees to make day supervisor and four employees to make night
supervisor. How many different ways can the two supervisors be hired?
Q3. In a club there are 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many
different possibilities are there?
Q4. A little league baseball team has six outfielders, seven infielders, five pitchers, and two catchers. Each
outfielder can play any of the three outfield positions and each infielder can play any of the four infield
positions. In how many ways can a team of nine players be chosen?
Q5. The baseball club has 50 members, 20 who are girls. A special group of 7 is to be selected. How many
ways can this group be selected if
A. everyone is considered equally
B. it must contain 4 girls
Q6. In a parent-teacher association, there are 7 parents and 5 teachers. A committee of 5 members is to be
chosen.
A. Find the number of committees have 3 parents and 2 teachers?
B. How many committees have more than 3 parents?
Q7. A box contains 24 transistors, 4 of which are defective. If 4 are sold at random, find the number of
possibilities of the following:
A. Exactly 2 are defective.
B. None is defective.
C. All are defective.
D. At least one is defective.