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Counting Problems

1) In how many ways can two married couples attending a concert be seated in a row of four seats if
a) Each married couple is seated together?
b) The members of each sex are seated together?

2) At a college library exhibition of faculty publications, four mathematics books, four social science books, and two biology books will be displayed on a shelf. (Assume that none of the books are alike.)
a) In how many ways can the ten books be arranged on the shelf?
b) In how many ways can the ten books be arranged on the shelf if books on the same subject matter are placed together?

3) A student planning her curriculum for the upcoming year must select one of three business courses, one of three mathematics courses, two of twelve elective courses, and either one of four history courses or one of three social science courses. How many different curricula are available for her consideration?

4) Find the number of distinguishable arrangements of each of the following "words."
a) acdbes
b) baaabe
c) aaabba

5) From a shipment of 40 transistors, 6 of which are defective, a sample of 8 transistors is selected at random.
a) In how many different ways can the sample be selected?
b) How many samples contain exactly 3 defective transistors?
c) How many samples contain no defective transistors?
d) How many samples contain at least 5 defective transistors?

Solution Preview

1a) There are 2 couples. The two couples can switch positions. The two members of each couple can switch positions.
2 * 2* 2 = 8
b) Same reason as in part (a), now each couple consists of the two members of the same sex.
2 * 2* 2 = 8

2a) Since the books are all different # of ways = 10! = ...

Solution Summary

Step-by-step solutions to all the problems are given.