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
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?
180
18
15
1500
150
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?
90
1765
24360
900
2876
24360
It is said that the three numbers are different. So the number of lock combinations is 30P3 = 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 5-card euchre hands have at least 2 black cards?
792
14000
5940
8731
35772
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?
1.3 x 1012
225
156900
11 × 1012
100875
1.3 x 1012
15P15 = 15!
= 1.307 674 368 × 1012
Example Question #1 : 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 + 26 + 26 + 9 + 9 + 9
26 x 26 x 26 x 9 x 9 x 9
None of the answers are correct
26 + 25 + 24 + 9 + 8 + 7
26 x 25 x 24 x 9 x 8 x 7
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 : Permutation / Combination
How many ways can 10 people win a race if ribbons are given for first, second, and third places?
120
None of the answers are correct
360
720
540
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?
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 #7 : 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?
12
5
1
7
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 #8 : Permutation / Combination
How many different 5 letter computer passwords are possible, assuming that letters cannot be repeated?
5,100,480
11,881,376
7,893,600
12,569,212
7,893,600
26P5 = 26 x 25 x 24 x 23 x 22 = 7,893,600
Example Question #1711 : Act Math
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.
36
24
60
48
72
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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