Permutation / Combination
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ACT Math › Permutation / Combination
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
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
How many different ways can five books be lined up on a shelf?
100
80
60
120
150
Explanation
Order matters, so we use permutations: (5)(4)(3)(2)(1) = 120
There are five possibilities for the first book, four possibilities for the second book, three for the third, and two for the fourth, and one possibility for the last book.
How many different ways can five books be lined up on a shelf?
100
80
60
120
150
Explanation
Order matters, so we use permutations: (5)(4)(3)(2)(1) = 120
There are five possibilities for the first book, four possibilities for the second book, three for the third, and two for the fourth, and one possibility for the last book.
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
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
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.
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.