All ACT Math Resources
Example Questions
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?
150
180
1500
18
15
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?
1765
900
2876
90
24360
24360
It is said that the three numbers are different. So the number of lock combinations is _{30}P_{3} = 24 360.
Example Question #1 : Permutation / Combination
The game of euchre uses the 9s, 10s, jacks, queens, kings, and aces from a standard deck of 52 cards. How many 5card euchre hands have at least 2 black cards?
35772
14000
5940
8731
792
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 #4 : 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?
100875
1.3 x 10^{12}
156900
225
11 × 10^{12}
1.3 x 10^{12}
_{15}P_{15} = 15!
= 1.307 674 368 × 10^{12}
Example Question #2 : 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?
26 x 26 x 26 x 9 x 9 x 9
26 + 25 + 24 + 9 + 8 + 7
26 + 26 + 26 + 9 + 9 + 9
26 x 25 x 24 x 9 x 8 x 7
None of the answers are correct
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 #131 : Integers
How many ways can 10 people win a race if ribbons are given for first, second, and third places?
None of the answers are correct
540
360
120
720
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 #4 : 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?
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
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 

36
3
7
12
12
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 #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?
1
7
12
5
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?
11,881,376
5,100,480
12,569,212
7,893,600
7,893,600
26P5 = 26 x 25 x 24 x 23 x 22 = 7,893,600