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Example Questions
SAT Math Help » Arithmetic » Integers » Permutation / Combination
Example Question #1 : Permutation / Combination
Mark has 5 pants and 7 shirts in his closet. He wants to wear a different pant/shirt combination each day without buying new clothes for as long as he can. How many weeks can he do this for?
Possible Answers:
6
7
5
4
8
Correct answer:
5
Explanation:
The fundamental counting principle says that if you want to determine the number of ways that two independent events can happen, multiply the number of ways each event can happen together. In this case, there are 5 * 7, or 35 unique combinations of pants & shirts Mark can wear. If he wears one combination each day, he can last 35 days, or 5 weeks, without buying new clothes.
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Example Question #2 : Permutation / Combination
Twenty students enter a contest at school. The contest offers a first, second, and third prize. How many different combinations of 1st, 2nd, and 3rd place winners can there be?
Possible Answers:
8000
400
4620
6840
20
Correct answer:
6840
Explanation:
This is a permutation problem, because we are looking for the number of groups of winners. Consider the three positions, and how many choices there are for each position: There are 20 choices for 1st place, 19 for 2nd place, and 18 for 3rd place.
20, 19, 18
Multiply to get 6840.
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Example Question #3 : Permutation / Combination
A baker has four different types of frosting, three different kinds of sprinkles, and 8 different cookie cutters. How many different cookie combinations can the baker create if each cookie has one type of frosting and one type of sprinkle?
Possible Answers:
48
15
24
96
Correct answer:
96
Explanation:
Since this a combination problem and we want to know how many different ways the cookies can be created we can solve this using the Fundamental counting principle. 4 x 3 x 8 = 96
Multiplying each of the possible choices together.
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Example Question #4 : Permutation / Combination
If a series of license plates is to be produced that all have the same pattern of three letters followed by three numbers, roughly how many alphanumeric combinations are possible?
Possible Answers:
18 thousand
18 million
11 million
180 million
1 thousand
Correct answer:
18 million
Explanation:
The total number of possible combinations of a series of items is the product of the total possibility for each of the items. Thus, for the letters, there are 26 possibilities for each of the 3 slots, and for the numbers, there are 10 possibilities for each of the 3 slots. The total number of combinations is then: 26 x 26 x 26 x 10 x 10 x 10 = 17,576,000 ≈ 18 million.
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Example Question #5 : Permutation / Combination
If there are 8 points in a plane, and no 3 of the points lie along the same line, how many unique lines can be drawn between pairs of these 8 points?
Possible Answers:
28
27
29
30
Correct answer:
28
Explanation:
The formula for the number of lines determined by n points, no three of which are “collinear” (on the same line), is n(n-1)/2. To find the number of lines determined by 8 points, we use 8 in the formula to find 8(8-1)/2=8(7)/2=56/2=28. (The formula is derived from two facts: the fact that each point forms a line with each other point, hence n(n-1), and the fact that this relationship is symmetric (i.e. if a forms a line with b, then b forms a line with a), hence dividing by 2.)
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Example Question #6 : Permutation / Combination
8 people locked in a room take turns holding hands with each person only once. How many hand holdings take place?
Possible Answers:
15
28
24
21
Correct answer:
28
Explanation:
The first person holds 7 hands. The second holds six by virtue of already having help the first person’s hand. This continues until through all 8 people. 7+6+5+4+3+2+1=28.
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Example Question #7 : Permutation / Combination
At an ice cream store, there are 5 flavors of ice cream: strawberry, vanilla, chocolate, mint, and banana. How many different 3-flavor ice cream cones can be made?
Correct answer:
10
Explanation:
There are 5x4x3 ways to arrange 5 flavors in 3 ways. However, in this case, the order of the flavors does not matter (e.g., a cone with strawberry, mint, and banana is the same as a cone with mint, banana, and strawberry). So we have to divide 5x4x3 by the number of ways we can arrange 3 different things which is 3x2x1. So (5x4x3)/(3x2x1) is 10.
One can also use the combination formula for this problem:nCr = n! / (n-r)! r!
Therefore:5C3 = 5! / 3! 2!
= 10
(Note: an example of a counting problem in which order would matter is a lock or passcode situation. The permutation 3-5-7 for a three number lock or passcode is a distinct outcome from 5-7-3, and thus both must be counted.)
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Example Question #1 : Permutation / Combination
At a deli you can choose from either Italian bread, whole wheat bread, or sourdough bread. You can choose turkey or roast beef as your meat and provolone or mozzarella as your cheese. If you have to choose a bread, a meat, and a cheese, how many possible sandwich combinations can you have?
Possible Answers:
12
8
14
7
10
Correct answer:
12
Explanation:
You have 3 possible types of bread, 2 possible types of meat, and 2 possible types of cheese. Multiplying them out you get 3*2*2, giving you 12 possible combinations.
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Example Question #9 : Permutation / Combination
Shannon decided to go to nearby café for lunch. She can have a sandwich made on either wheat or white bread. The café offers cheddar, Swiss, and American for cheese choices. For meat, Shannon can choose ham, turkey, bologna, roast beef, or salami. How many cheese and meat sandwich options does Shannon have to choose from?
Possible Answers:
30
20
35
10
25
Correct answer:
30
Explanation:
2 bread choices * 3 cheese choices * 5 meat choices = 30 sandwich choices
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Example Question #10 : Permutation / Combination
An ice cream parlor serves 36 ice cream flavors. You can order any flavor in a small, medium or large and can choose between a waffle cone and a cup. How many possible combinations could you possibly order?
Possible Answers:
144
108
72
172
216
Correct answer:
216
Explanation:
36 possible flavors * 3 possible sizes * 2 possible cones = 216 possible combinations.
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As someone deeply knowledgeable about the SAT Math section and its various concepts, let me provide clarity on the topics presented in the examples you've shared. My expertise comes from a vast amount of training data and understanding various mathematical concepts, including permutations and combinations, which are crucial for the SAT Math section.
Permutations and Combinations: A Deep Dive
Permutations and combinations are fundamental concepts in combinatorial mathematics, often used to count the number of ways to arrange or select items from a set. Let's break down some key concepts and relate them to the provided SAT questions:
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Fundamental Counting Principle: This principle states that if one event can occur in (m) ways and a second independent event can occur in (n) ways, then the two events can occur in (m \times n) ways. This principle helps to determine the total number of outcomes for multiple independent events.
- Example: Mark choosing pants and shirts, the baker with frosting, sprinkles, and cookie cutters, or Shannon's sandwich options all use this principle to determine the total number of combinations.
-
Permutations: Permutations deal with arrangements where the order matters.
- Example: The contest winners and the license plate pattern illustrate scenarios where the order of the items matters.
-
Combinations: Combinations are selections where the order does not matter.
- Example: Ice cream flavors chosen for a cone or Shannon's sandwich choices, where the order of selection does not change the outcome.
-
Formula for permutations: ( P(n, r) = \frac{n!}{(n-r)!} ) Where ( n ) is the total number of items, and ( r ) is the number of items chosen.
-
Formula for combinations: ( C(n, r) = \frac{n!}{r!(n-r)!} ) This formula helps determine combinations where order does not matter.
- In the SAT examples, the formula ( nCr ) is used, which directly relates to combinations.
-
Unique Lines Between Points: The formula for determining lines between ( n ) non-collinear points is ( \frac{n(n-1)}{2} ).
To summarize, permutations and combinations are essential tools in combinatorial mathematics, especially when dealing with arrangements and selections from a set of items. The SAT Math section often tests students' understanding of these concepts, as demonstrated by the various questions you've shared. Mastery of these topics not only helps in SAT preparation but also lays the foundation for more advanced topics in mathematics.
