CSCE 235

Handout 14b: Counting Exercises – Solution

Assigned March 10, 2002

The Addition And Multiplication Rules

One of the difficult issues in understanding the addition and multiplication rules is to know when a rule is applicable. We know from Handout 13 that:

The number of ways in which precisely one of a collection of mutually exclusive events can occur is the sum of the number of ways in which each event can occur (basically the above). And this is when the addition rule is applicable.

The number of ways in which a sequence of events can occur is the product of the number of ways in which each individual event can occur. And this is when the multiplication rule is applicable.

To learn how to recognize the solution approach to a problem, you need to clearly know what the problem is asking for and also you need to practice.

So, this handout has a list of problems. Identify whether the addition rule should be used (A), or the multiplication rule should be used (M), or both (A+M) to solve each of the following problems. You do not need to solve the problems. I will post the answers (A, M, or A+M) on the class’ website (under Handouts).

1. From a group of 13 men, 6 women, 2 boys, and 4 girls,

(a) In how many ways can a man, a woman, a boy, and a girl be selected? M

(b) In how many ways can a man or a girl be selected? A

(c) In how many ways can one person be selected? A

2. Using only the digits 1, 3, 5, 7, and 9,

(a) How many two-digit numbers can be formed? M

(b) How many three-digit numbers can be formed? M

(c) How many four-digit numbers can be formed? M

(d) How many two- or three- or four-digit numbers can be formed? A+M

3. A building supplies store carries metal, wood, and plastic moldings. Metal and wood molding comes in two different colors. Plastic molding comes in six different colors.

(a) How many choices of molding does this store offer? A

(b) If each kind and each color of molding comes in four different lengths, how many choices does the consumer have in the purchase of one piece of molding? A

4. (a) In how many ways can two dice land? M

(b) In how many ways can five dice land? M

(c) In how many ways can n dice land? M

5. A standard deck of 52 playing cards has four suits (spades, hearts, clubs, and diamonds). Each suit has 13 cards, from 2 to 10 and J, Q, K, and Ace. In how many ways can one draw from a standard deck of 52 playing cards

(a) A heart or a spade? A

(b) An ace or a king? A

(c) A card numbered 2 through 10? A

(d) A card numbered 2 through 10 or a king? A

(e) Two cards: the first card an ace of a king; and then after replacing the first card into the deck, the second card an ace or a king again? A+M

6. Suppose are three different routes from the UNL City Campus to the UNL East Campus, and there are five different routes from the UNL East Campus to the Lincoln Municipal Airport.

(a) How many different routes are there from the UNL City Campus to the Airport via the UNL East Campus (say, you need to pick up your friends at the UNL East Campus)? M

(b) How many different round trips are there from the Airport to the UNL City Campus, passing through the UNL East Campus each way? M

(c) Repeat (b) but with another constraint: you don’t want to go through the same route more than once. A+M

7. Let be a set of n elements and .

(a) How many functions are there from A to B? A+M

(b) How many onto functions are there from A to B? A+M

8. (a) In how many ways can the letters a, b, c, d, e, and f be arranged so that the letters a and b are adjacent? A+M

(b) In how many ways can the letters a, b, c, d, e, and f be arranged so that the letters a and b are not adjacent? A+M

Based on (Ross and Wright 1988) and (Goodaire and Parmenter 2002).