104 TECHNIQUES OF COUNTING [CHAP. 5
PERMUTATIONS WITH REPETITIONS, ORDERED SAMPLES
5.51. Find the number of permutations that can be formed from all the letters of each word: (a) QUEUE; (b) COMMITTEE;
(c) PROPOSITION; (d) BASEBALL.
5.52. Suppose we are given 4 identical red flags, 2 identical blue flags, and 3 identical green flags. Find the numbermof
different signals that can be formed by hanging the 9 flags in a vertical line.
5.53. A box contains 12 lightbulbs. Find the numbernof ordered samples of size 3:
(a) with replacement; (b) without replacement.
5.54. A class contains l0 students. Find the numbernof ordered samples of size 4:
(a) with replacement; (b) without replacement.COMBINATIONS
5.55. A restaurant has 6 different desserts. Find the number of ways a customer can choose:
(a) 1 dessert; (b) 2 of the desserts; (c) 3 of the desserts.
5.56. A class contains 9 men and 3 women. Find the number of ways a teacher can select a committee of 4 from the class
where there is:
(a) no restrictions; (b) 2 men and 2 women; (c) exactly one woman; (d) at least one woman.
5.57. A woman has 11 close friends. Find the number of ways she can invite 5 of them to dinner where:
(a) There are no restrictions.
(b) Two of the friends are married to each other and will not attend separately.
(c) Two of the friends are not speaking with each other and will not attend together.
5.58. Aclass contains 8 men and 6 women and there is one married couple in the class. Find the numbermof ways a teacher
can select a committee of 4 from the class where the husband or wife but not both can be on the committee.
5.59. A box has 6 blue socks and 4 white socks. Find the number of ways two socks can be drawn from the box where:
(a) There are no restrictions. (b) They are different colors. (c) They are the same color.
5.60. A women student is to answer 10 out of 13 questions. Find the number of her choices where she must answer:
(a) the first two questions; (c) exactly 3 out of the first 5 questions;
(b) the first or second question but not both; (d) at least 3 of the first 5 questions.INCLUSION–EXCLUSION PRINCIPLE
5.61. Suppose 32 students are in an art classAand 24 students are in a biology classB, and suppose 10 students are in both
classes. Find the number of students who are:
(a) in classAor in classB; (b) only in classA; (c) only in classB.
5.62. A survey of 80 car owners shows that 24 own a foreign-made car and 60 own a domestic-made car. Find the number
of them who own:
(a) both a foreign made car and a domestic made car;
(b) only a foreign made car;
(c) only a domestic made car.
5.63. Consider all integers from 1 up to and including 100. Find the number of them that are:
(a) odd or the square of an integer; (b) even or the cube of an integer.
5.64. In a class of 30 students, 10 gotAon the first test, 9 gotAon a second test, and 15 did not get anAon either test.
Find: the number of students who got:(a) anAon both tests;
(b) anAon the first test but not the second;
(c) anAon the second test but not the first.5.65. Consider all integers from 1 up to and including 300. Find the number of them that are divisible by:
(a) at least one of 3, 5, 7; (c) by 5, but by neither 3 nor 7;
(b) 3 and 5 but not by 7; (d) by none of the numbers 3, 5, 7.