Permutations and Combinations Matching Worksheet for math practice.

Problem Analysis:

The worksheet involves solving problems related to permutations and combinations. Each problem requires identifying whether the scenario involves permutations (order matters) or combinations (order does not matter). Let's solve each problem step by step.

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Problem 1: How many different ways can you fill 1st, 2nd, and 3rd places with 10 runners?

- Key Concept: This is a permutation problem because the order of placing (1st, 2nd, 3rd) matters.
- 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 to choose.
- Here, \( n = 10 \) (runners) and \( r = 3 \) (places to fill).
- Calculation:
\[
P(10, 3) = \frac{10!}{(10-3)!} = \frac{10!}{7!} = 10 \times 9 \times 8 = 720
\]
- Answer: \( 720 \)

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Problem 2: How many different combinations of management can there be to fill the positions of manager, secretary, and administrative assistant, assuming only one person can hold one position?

- Key Concept: This is a permutation problem because the order of assigning roles (manager, secretary, admin assistant) matters.
- Formula for Permutations: \( P(n, r) = \frac{n!}{(n-r)!} \), where \( n \) is the total number of candidates, and \( r \) is the number of positions.
- Here, \( n = 10 \) (candidates) and \( r = 3 \) (positions).
- Calculation:
\[
P(10, 3) = \frac{10!}{(10-3)!} = \frac{10!}{7!} = 10 \times 9 \times 8 = 720
\]
- Answer: \( 720 \)

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Problem 3: How many different words can we make using the letters in "CAT"?

- Key Concept: This is a permutation problem because the order of letters matters when forming words.
- Formula for Permutations of Distinct Letters: \( n! \), where \( n \) is the number of distinct letters.
- Here, the word "CAT" has 3 distinct letters.
- Calculation:
\[
3! = 3 \times 2 \times 1 = 6
\]
- Answer: \( 6 \)

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Problem 4: Robin has 5 different pairs of shoes that match with 6 different pairs of socks. How many ways can she select one pair of shoes and one pair of socks?

- Key Concept: This is a basic counting principle problem (multiplication rule).
- Formula: If there are \( m \) ways to do one thing and \( n \) ways to do another, then there are \( m \times n \) ways to do both.
- Here, Robin has 5 choices for shoes and 6 choices for socks.
- Calculation:
\[
5 \times 6 = 30
\]
- Answer: \( 30 \)

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Problem 5: Ginny has 6 water colors (red, green, blue, yellow, black, white). In how many different orders can all 6 different water color appear?

- Key Concept: This is a permutation problem because the order of arranging the colors matters.
- Formula for Permutations of All Items: \( n! \), where \( n \) is the total number of items.
- Here, \( n = 6 \) (colors).
- Calculation:
\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\]
- Answer: \( 720 \)

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Problem 6: How many different sandwiches can be made with 6 packets of bread, 4 pieces of cheese, and 3 packets of ham (assuming one of each is used)?

- Key Concept: This is a basic counting principle problem (multiplication rule).
- Formula: If there are \( m \) ways to choose bread, \( n \) ways to choose cheese, and \( p \) ways to choose ham, then there are \( m \times n \times p \) ways to make a sandwich.
- Here, there are 6 choices for bread, 4 choices for cheese, and 3 choices for ham.
- Calculation:
\[
6 \times 4 \times 3 = 72
\]
- Answer: \( 72 \)

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Problem 7: How many 2-digit numbers can you make using the digits 1, 2, 3, and 4 without repeating the digits?

- Key Concept: This is a permutation problem because the order of digits matters.
- Formula for Permutations: \( P(n, r) = \frac{n!}{(n-r)!} \), where \( n \) is the total number of digits, and \( r \) is the number of digits to choose.
- Here, \( n = 4 \) (digits: 1, 2, 3, 4) and \( r = 2 \) (2-digit numbers).
- Calculation:
\[
P(4, 2) = \frac{4!}{(4-2)!} = \frac{4!}{2!} = 4 \times 3 = 12
\]
- Answer: \( 12 \)

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Problem 8: Eva has 4 glasses of water. In how many different orders can the glasses of water be arranged?

- Key Concept: This is a permutation problem because the order of arranging the glasses matters.
- Formula for Permutations of All Items: \( n! \), where \( n \) is the total number of items.
- Here, \( n = 4 \) (glasses).
- Calculation:
\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]
- Answer: \( 24 \)

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Final Answers:

1. \( 720 \)
2. \( 720 \)
3. \( 6 \)
4. \( 30 \)
5. \( 720 \)
6. \( 72 \)
7. \( 12 \)
8. \( 24 \)

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