### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Find The Greatest Or Least Number Of Combinations

A candy shop sells Valentine's Day gift baskets that consist of chocolates, a basket, and a card. If there are five different types of chocolate, three types of baskets, and ten options for cards, how many different gift basket combinations are there?

**Possible Answers:**

1500

15

150

180

18

**Correct answer:**

150

The correct answer is 150. Since there are five types of chocolate, three types of baskets, and ten card choices, the correct answer can be found by multiplying 3 x 5 x 10, which is 150. If you got 15 or 1500, you may have made a multiplication error. If you got 18, you may have added instead of multiplying.

### Example Question #2 : Permutation / Combination

A locker combination consists of three different numbers from the set of 30 different numbers on the face of the lock. Imagine that you have forgotten the combination. How many times do you have to try to find the right combination?

**Possible Answers:**

1765

2876

24360

90

900

**Correct answer:**

24360

It is said that the three numbers are different. So the number of lock combinations is _{30}*P*_{3} = 24 360.

### Example Question #3 : Permutation / Combination

The game of euchre uses the 9s, 10s, jacks, queens, kings, and aces from a standard deck of 52 cards. How many 5-card euchre hands have at least 2 black cards?

**Possible Answers:**

14000

35772

792

8731

5940

**Correct answer:**

35772

The hand could have 2, 3, 4, or 5 black cards. There are 12 black cards and 12 red cards, so the numbers of combinations for the four cases are as follows.

2 black cards: *C*(12, 2) × *C*(12, 3) = 14 520

3 black cards: *C*(12, 3) × *C*(12, 2) = 14 520

4 black cards: *C*(12, 4) × *C*(12, 1) = 5940

5 black cards: *C*(12, 5) × *C*(12, 0) = 792

The total number of euchre hands that have at least two black cards is the total of these four cases, 35 772.

### Example Question #1 : Permutation / Combination

You work as a health inspector and must visit each of the 15 restaurants in town once each week. In how many different orders can you make these inspections?

**Possible Answers:**

156900

225

100875

1.3 x 10^{12}

11 × 10^{12}

**Correct answer:**

1.3 x 10^{12}

_{15}*P*_{15} = 15!

= 1.307 674 368 × 10^{12}

^{ }

### Example Question #5 : Permutation / Combination

A license plate consists of three letters followed by three numbers (excluding 0). How many license plates can be made if no letters or numbers are repeated?

**Possible Answers:**

26 + 25 + 24 + 9 + 8 + 7

26 x 26 x 26 x 9 x 9 x 9

26 + 26 + 26 + 9 + 9 + 9

26 x 25 x 24 x 9 x 8 x 7

None of the answers are correct

**Correct answer:**

26 x 25 x 24 x 9 x 8 x 7

There are 26 letters in the alphabet and 9 digits when you exclude 0. Each selection can go with any other selection, so each number is multiplied together. After the first letter is picked, the sample size (what you can pick from) is reduced by one because there is no repetition. So the answer 26 x 25 x 24 x 9 x 8 x 7 is correct. If repetition were allowed answer 26 x 26 x 26 x 9 x 9 x 9 would be correct.

### Example Question #1 : How To Find The Greatest Or Least Number Of Combinations

How many ways can 10 people win a race if ribbons are given for first, second, and third places?

**Possible Answers:**

720

120

360

None of the answers are correct

540

**Correct answer:**

720

Independent events are multiplied. Once the first place is chosen, the sample space (what you can pick from) is reduced by one since there is no repetition (you can’t win first and second places at the same time). Thus, 10 x 9 x 8 = 720

### Example Question #1 : Permutation / Combination

Sam is getting dressed in the morning and has 6 pairs of pants, 4 shirts, and 5 pairs of socks to choose from. How many distinct combinations consisting of 1 pair of pants, 1 shirt and 1 pair of socks can Sam make?

**Possible Answers:**

**Correct answer:**

In order to find the answer, multiply the quantities together:

This is because for each pair of pants, there are 4 options for shirts and 5 options for socks.

### Example Question #1 : Permutation / Combination

Susie wants to make a sandwich for lunch. She has two types of breads, three types of meats, and two types of cheeses to choose from. How many different sandwiches can she make if she chooses only one of each ingredient?

**Possible Answers:**

1

7

5

12

**Correct answer:**

12

Each item (bread, meat, and cheese) is chosen independently from the others, so the answer can be found in a tree diagram: Bread x Meat x Cheese or 2 x 3 x 2 or 12.

### Example Question #1 : Permutation / Combination

How many different 5 letter computer passwords are possible, assuming that letters cannot be repeated?

**Possible Answers:**

12,569,212

7,893,600

5,100,480

11,881,376

**Correct answer:**

7,893,600

26P5 = 26 x 25 x 24 x 23 x 22 = 7,893,600

### Example Question #1 : Permutation / Combination

At a party with 9 guests, every guest shakes every other guest’s hand exactly once. How many handshakes are exchanged during the party?

When two people shake hands with each other, that counts as one handshake.

**Possible Answers:**

24

72

60

36

48

**Correct answer:**

36

Each person shakes 8 people’s hands, so at first guess that’s 9x8=72 handshakes. However, this double counts the number of handshakes since we count the handshake between person A and B once when we count A’s 8 handshakes and a second time when we count B’s 8 handshakes. Therefore, we divide our estimate by 2 and get 36.

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