ACT Math : How to find the greatest or least number of combinations

Study concepts, example questions & explanations for ACT Math

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Example Questions

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Example Question #1 : Permutation / Combination

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:

15

150

18

1500

180

Correct answer:

150

Explanation:

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

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:

90

1765

900

2876

24360

Correct answer:

24360

Explanation:

It is said that the three numbers are different. So the number of lock combinations is 30P3 = 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:

5940

14000

792

35772

8731

Correct answer:

35772

Explanation:

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 #2 : 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:

100875

156900

1.3 x 1012

225

11 × 1012

Correct answer:

1.3 x 1012

Explanation:

15P15 = 15!

= 1.307 674 368 × 1012

 

Example Question #2 : How To Find The Greatest Or Least Number Of Combinations

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

None of the answers are correct

26 + 26 + 26 + 9 + 9 + 9

26 x 26 x 26 x 9 x 9 x 9

26 x 25 x 24 x 9 x 8 x 7

Correct answer:

26 x 25 x 24 x 9 x 8 x 7

Explanation:

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 #1715 : Act Math

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

Possible Answers:

720

540

120

360

None of the answers are correct

Correct answer:

720

Explanation:

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 #5 : 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:

\dpi{100} \small 16

\dpi{100} \small 26

\dpi{100} \small 144

\dpi{100} \small 220

\dpi{100} \small 120

Correct answer:

\dpi{100} \small 120

Explanation:

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

\dpi{100} \small (6)(4)(5) = 120

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

Example Question #2 : How To Find The Greatest Or Least Number Of Combinations

Sally is putting on jewelry and has decided to wear one necklace, one pair of earrings, and one ring. Her jewelry collection is listed below. How many different combinations of jewelry can she wear?

 

Necklace

Earrings

Ring

short

studs

gold

long

hoops

silver

 

dangling

 

 

 

 

 

 

Possible Answers:

7

3

36

12

Correct answer:

12

Explanation:

To find the number of different combinations, we must use the fundamental counting principal to multiply the number of options in each category together:

(2)(3)(2) = 12

 

 

Example Question #4 : 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

12

5

7

Correct answer:

12

Explanation:

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 #2 : How To Find The Greatest Or Least Number Of Combinations

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

 

Possible Answers:

12,569,212

7,893,600

11,881,376

5,100,480

Correct answer:

7,893,600

Explanation:

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

 

 

 

 

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