How to find the greatest or least number of combinations
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ACT Math › How to find the greatest or least number of combinations
The menu above is from Lena’s Italian Kitchen. If you are going there for dinner, how many different combinations of a meal and a salad are there?
20
5
10
15
25
Explanation
Combinations = # first option * # second option
= # meals * # salads
= 5 * 4 = 20
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 |
7
3
12
36
18
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
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
225
11 × 1012
156900
1.3 x 1012
Explanation
15_P_15 = 15!
= 1.307 674 368 × 1012
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.
24
36
48
60
72
Explanation
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.
How many ways can 10 people win a race if ribbons are given for first, second, and third places?
120
360
540
720
None of the answers are correct
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
How many different 5 letter computer passwords are possible, assuming that letters cannot be repeated?
7,893,600
5,100,480
11,881,376
12,569,212
Explanation
26P5 = 26 x 25 x 24 x 23 x 22 = 7,893,600
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
18
1500
15
180
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.
A student is taking a test consisting of six questions. It is a multiple choice test and each question has four answers labelled A, B, C, and D. How many ways can the student answer all six questions if he does not choose the same answer for any two consecutive questions?
972
625
324
100
30
Explanation
the student has four choices for the first question, but only three choices for each of the remaining questions since he does not choose answers with the same letter twice in a row. From the multiplicative counting principle, there are 4 × 3 × 3 × 3 × 3 x 3 = 972 ways Steve can answer the six questions.
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
24360
2876
1765
900
Explanation
It is said that the three numbers are different. So the number of lock combinations is 30_P_3 = 24 360.
A university can send 3 track athletes and 2 field athletes to an upcoming sports event. The university has 14 track athletes and 10 field athletes who qualify. How many different teams can be chosen to attend the sports event?
10,080
16,380
64,800
840
140
Explanation
The number of different teams that could be chosen is C(14, 3) × C(10, 2) = 16,380.