NEL 90 4.1 Counting Principles

4.1

Example 1

Kim is choosing a new cellphone. She can choose the 300, 400, or 500 model, in green, pink, indigo, orange, or taupe trim. Kim is also deciding whether or not to get a data plan for her phone.

a) How many different choices does Kim have altogether, accounting for model, colour, and data plan or not?

b) The store agent advises Kim not to choose the 300 model if she wants a data plan. Also, the 500 model is not available in indigo or taupe trim. How many choices does Kim have now?

Solution

a) Kim must choose a model AND a colour AND whether to get a data plan. I applied

the Fundamental Counting Principle.

Number of ways to choose a model: a 5 3

Number of ways to choose a colour: b 5 5

Number of ways to choose a data plan or not: c 5 2

Total number of ways to choose a phone: a ? b ? c 5 3 ? 5 ? 2, or 30 ways

Kim has 30 different choices.

Counting Principles

Keep in Mind

The Fundamental Counting Principle says the following:

• If one task can be performed in a ways AND another task can be performed in b ways, then both tasks can be performed in a ? b ways.

• If a series of tasks can be performed in a ways AND b ways AND c ways and so on, then all these tasks can be performed in a ? b ? c … ways.

For OR situations, the Fundamental Counting Principle does not apply:

• If the tasks are mutually exclusive, they involve disjoint sets A and B, and n(A c B) 5 n(A) 1 n(B).

• If the tasks are not mutually exclusive, they involve non-disjoint sets C and D, and n(C c D) 5 n(C) 1 n(D) 2 n(C d D) (the Principle of Inclusion and Exclusion).

Outcome tables, organized lists, and tree diagrams are useful to help solve counting problems when you need to display all the possible outcomes.

YOU WILL NEED• calculator

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91NEL 4.1 Counting Principles

b) This is a more complicated situation, so I used a tree diagram.

When there are more than two tasks, a tree diagram is a better strategy than an

outcome table.

There are 21 branches, so Kim now has 21 ways to choose a phone.

Example 2

From a standard deck of cards, how many possibilities are there for drawing a) a two or a face card? b) a spade or a queen?

Solution

Step 1. I recognized that both parts of the problem were OR situations.

Step 2. I let A represent the event of drawing a two and B represent the event of drawing

a face card (jacks, queens, and kings). I knew that a card could either be a two or a

face card, but not both, so the events are mutually exclusive.

n(A c B) 5 n(A) 1 n(B)

n(A c B) 5 4 1 12, or 16

a) There are 16 possibilities for drawing a two or a face card.

Step 3. I let S represent the event of drawing a spade, Q represent the event of drawing a

queen, and U represent all possible draws of one card from a standard deck. I drew a

Venn diagram to visualize how the two events are related.

Step 4. I could see that I needed the number of cards in the union of the two sets.

I applied the Principle of Inclusion and Exclusion, to make sure I did not count the

queen of spades twice.

n 1S c Q 2 5 n 1S 2 1 n 1Q 2 2 n 1S d Q 2 n 1S c Q 2 5 13 1 4 2 1, or 16

b) There are 16 possibilities for drawing a spade or a queen.

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92 NEL 4.1 Counting Principles

Practice

1. Bryce likes to vary the way he takes his coffee. He can have it black, with milk, or with cream, and he can also decide whether or not to have sugar.

a) Create an outcome table to count the total number of ways Bryce can take his coffee.

b) Use the Fundamental Counting Principle to verify your response to part a).

2. Marie and Sitka are in charge of repainting their homeroom. They have a choice of mauve, lime green, or pale gold paint, with either a gloss or a matte finish, and they can go with either an ivory or a cream trim.

a) Complete this tree diagram to count all of Marie and Sitka’s choices.

b) Use the Fundamental Counting Principle to verify your response to part a).

3. For each of the following situations, state whether the Fundamental Counting Principle applies. Explain how you know.

a) choosing either a keypad or touch-screen phone from one of 4 providers, with or without a data plan

b) drawing either a red card or a face card from a standard deck of cards

c) choosing a pizza with regular, deep, or thin crust; pepperoni, mushroom, or ham topping; and mozzarella or cheddar cheese

d) rolling a sum of 8 on two standard dice

4. Count the number of possibilities for each situation in question 3.

a) c)

b) d)

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5. A combination lock uses the letters A through H on the first dial and the digits 0 through 9 on the second and third dials.

a) How many possible combinations are there for this lock?

b) How many combinations do not use the same digit twice?

6. a) How many ways are there to draw either a red number card (2 through 10) or a face card from a standard deck of cards?

b) How many ways are there to draw either a red number card or a 7?

7. A set of cards contains 8 triangle cards, 5 circle cards, 7 kite cards, and 3 rectangle cards. Four of the cards are shown. If Peter draws one card from the whole set, how many possibilities are there? Explain how you know.

MULTIPLE CHOICE

8. In a set of 13 distinct natural numbers, 3 of the numbers are prime numbers, and 4 of them are even. The number 2 is in the set. Sondra wants to choose an odd prime number from the set. Which of the following applies?

A. the Fundamental Counting Principle

B. the Principle of Inclusion and Exclusion

C. both principles

D. neither principle

WRITTEN RESPONSE

9. Misha is ordering a customized salad from a deli counter. He can create his salad from the choices on the right.

a) How many different salads could Misha create altogether? Show your reasoning.

b) Misha does not like cucumber. Also, he has decided to have either croutons or bacon bits but not both. How many different salads could he create to meet these requirements? Show your reasoning.

• Boston, romaine, or baby spinach leaves

• cucumber or tomato but not both • croutons, bacon bits, both, or neither • ranch, thousand island, balsamic, or

French dressing

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