permutation or combination?
directions: recall that a permutation is an arrangement with a specific order, while a combination is a group with no specific order. determine whether the example represents a permutation or combination, then solve.

| example | p or c? | work solution and answer |
| --- | --- | --- |
| 1) how many ways can 4 candy bars be chosen from a store that sells 30 candy bars? | | |
| 2) how many ways can 13 students line up for lunch? | | |
| 3) how many ways can you make a 3 - letter arrangements out of the letters in the word trapezoid. | | |
| 4) how many ways can you choose 2 books from a shelf of 40 books? | | |
| 5) how many ways can 12 swimmers finish in first, second, and third place? | | |
| 6) how many ways can mrs. sullivan choose two students from 27 to help put away calculators at the end of class? | | |
| 7) how many different 12 - letter arrangements can be made using the letters in the word indianapolis? | | |
| 8) how many different 11 - letter arrangements can be made using the letters in the word firecracker? | | |
| 9) there are 18 offensive players on the hockey team. how many ways can the coach choose a left wing, center, and right wing to start the game? | | |

Question

permutation or combination?
directions: recall that a permutation is an arrangement with a specific order, while a combination is a group with no specific order. determine whether the example represents a permutation or combination, then solve.

| example | p or c? | work solution and answer |
| --- | --- | --- |
| 1) how many ways can 4 candy bars be chosen from a store that sells 30 candy bars? |  |  |
| 2) how many ways can 13 students line up for lunch? |  |  |
| 3) how many ways can you make a 3 - letter arrangements out of the letters in the word trapezoid. |  |  |
| 4) how many ways can you choose 2 books from a shelf of 40 books? |  |  |
| 5) how many ways can 12 swimmers finish in first, second, and third place? |  |  |
| 6) how many ways can mrs. sullivan choose two students from 27 to help put away calculators at the end of class? |  |  |
| 7) how many different 12 - letter arrangements can be made using the letters in the word indianapolis? |  |  |
| 8) how many different 11 - letter arrangements can be made using the letters in the word firecracker? |  |  |
| 9) there are 18 offensive players on the hockey team. how many ways can the coach choose a left wing, center, and right wing to start the game? |  |  |

Accepted Answer

### Problem 1:
# Explanation:
## Step1: Identify P or C
Choosing candy bars, order doesn't matter, so combination (C).
## Step2: Apply combination formula
The formula for combinations is $ _nC_r=\frac{n!}{r!(n - r)!} $, where $ n = 30 $, $ r=4 $.
$ _{30}C_4=\frac{30!}{4!(30 - 4)!}=\frac{30!}{4!×26!}=\frac{30×29×28×27}{4×3×2×1} $
## Step3: Calculate the value
$ \frac{30×29×28×27}{24}=\frac{30×29×28×27}{24}=27405 $
# Answer:
P or C? : C; Number of ways: $ 27405 $

### Problem 2:
# Explanation:
## Step1: Identify P or C
Lining up students, order matters, so permutation (P).
## Step2: Apply permutation formula
The formula for permutations is $ _nP_r=\frac{n!}{(n - r)!} $, where $ n = 13 $, $ r = 13 $ (since we are lining up all 13 students).
$ _{13}P_{13}=\frac{13!}{(13 - 13)!}=\frac{13!}{0!}=13! $
## Step3: Calculate the value
$ 13!=13×12×11×\cdots×1 = 6227020800 $
# Answer:
P or C? : P; Number of ways: $ 6227020800 $

### Problem 3:
# Explanation:
## Step1: Identify P or C
Making letter arrangements, order matters, so permutation (P). First, find the number of distinct letters in "TRAPEZOID". The word "TRAPEZOID" has 9 distinct letters (T, R, A, P, E, Z, O, I, D). We are making 3 - letter arrangements, so $ n = 9 $, $ r=3 $.
## Step2: Apply permutation formula
Using $ _nP_r=\frac{n!}{(n - r)!} $, with $ n = 9 $, $ r = 3 $.
$ _{9}P_{3}=\frac{9!}{(9 - 3)!}=\frac{9!}{6!}=9×8×7 $
## Step3: Calculate the value
$ 9×8×7 = 504 $
# Answer:
P or C? : P; Number of ways: $ 504 $

### Problem 4:
# Explanation:
## Step1: Identify P or C
Choosing books, order doesn't matter, so combination (C).
## Step2: Apply combination formula
Using $ _nC_r=\frac{n!}{r!(n - r)!} $, where $ n = 40 $, $ r = 2 $.
$ _{40}C_2=\frac{40!}{2!(40 - 2)!}=\frac{40!}{2!×38!}=\frac{40×39}{2×1} $
## Step3: Calculate the value
$ \frac{40×39}{2}=780 $
# Answer:
P or C? : C; Number of ways: $ 780 $

### Problem 5:
# Explanation:
## Step1: Identify P or C
Finishing in first, second, third place, order matters, so permutation (P). Here $ n = 12 $, $ r=3 $.
## Step2: Apply permutation formula
Using $ _nP_r=\frac{n!}{(n - r)!} $, with $ n = 12 $, $ r = 3 $.
$ _{12}P_{3}=\frac{12!}{(12 - 3)!}=\frac{12!}{9!}=12×11×10 $
## Step3: Calculate the value
$ 12×11×10 = 1320 $
# Answer:
P or C? : P; Number of ways: $ 1320 $

### Problem 6:
# Explanation:
## Step1: Identify P or C
Choosing students to help, order doesn't matter, so combination (C). Here $ n = 27 $, $ r = 2 $.
## Step2: Apply combination formula
Using $ _nC_r=\frac{n!}{r!(n - r)!} $, with $ n = 27 $, $ r = 2 $.
$ _{27}C_2=\frac{27!}{2!(27 - 2)!}=\frac{27!}{2!×25!}=\frac{27×26}{2×1} $
## Step3: Calculate the value
$ \frac{27×26}{2}=351 $
# Answer:
P or C? : C; Number of ways: $ 351 $

### Problem 7:
# Explanation:
## Step1: Identify P or C
Arranging letters, order matters, so permutation. First, find the number of letters and repeated letters in "INDIANAPOLIS". The word "INDIANAPOLIS" has 12 letters with repeated letters: I: 3, N: 2, A: 2, P: 1, O: 1, L: 1, S: 1. The formula for permutations of multi - set is $ \frac{n!}{n_1!n_2!\cdots n_k!} $, where $ n = 12 $, $ n_1 = 3 $ (for I), $ n_2=2 $ (for N), $ n_3 = 2 $ (for A) and the rest $ n_i = 1 $.
## Step2: Apply multi - set permutation formula
$ \frac{12!}{3!×2!×2!}=\frac{12×11×10×9×8×7×6×5×4×3!}{3!×2×1×2×1}=\frac{12×11×10×9×8×7×6×5×4}{4} $
## Step3: Calculate the value
$ 12×11×10×9×8×7×6×5 = 19958400 $
# Answer:
P or C? : P; Number of arrangements: $ 19958400 $

### Problem 8:
# Explanation:
## Step1: Identify P or C
Arranging letters, order matters, so permutation. First, find the number of letters and repeated letters in "FIRECRACKER". The word "FIRECRACKER" has 11 letters with repeated letters: R: 3, E: 2, C: 2, F: 1, I: 1, A: 1, K: 1. The formula for permutations of multi - set is $ \frac{n!}{n_1!n_2!\cdots n_k!} $, where $ n = 11 $, $ n_1 = 3 $ (for R), $ n_2 = 2 $ (for E), $ n_3=2 $ (for C) and the rest $ n_i = 1 $.
## Step2: Apply multi - set permutation formula
$ \frac{11!}{3!×2!×2!}=\frac{11×10×9×8×7×6×5×4×3!}{3!×2×1×2×1}=\frac{11×10×9×8×7×6×5×4}{4} $
## Step3: Calculate the value
$ 11×10×9×8×7×6×5 = 1663200 $
# Answer:
P or C? : P; Number of arrangements: $ 1663200 $

### Problem 9:
# Explanation:
## Step1: Identify P or C
Choosing left wing, center, right wing, order matters (different positions), so permutation (P). Here $ n = 18 $, $ r = 3 $.
## Step2: Apply permutation formula
Using $ _nP_r=\frac{n!}{(n - r)!} $, with $ n = 18 $, $ r = 3 $.
$ _{18}P_{3}=\frac{18!}{(18 - 3)!}=\frac{18!}{15!}=18×17×16 $
## Step3: Calculate the value
$ 18×17×16 = 4896 $
# Answer:
P or C? : P; Number of ways: $ 4896 $

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QUESTION IMAGE

Question

Explanation:

Problem 1:

Step1: Identify P or C

Step2: Apply combination formula

Step3: Calculate the value

Step1: Identify P or C

Step2: Apply permutation formula

Step3: Calculate the value

Step1: Identify P or C

Step2: Apply permutation formula

Step3: Calculate the value

Answer:

Problem 2:

example	p or c?	work solution and answer
2) how many ways can 13 students line up for lunch?
3) how many ways can you make a 3 - letter arrangements out of the letters in the word trapezoid.
4) how many ways can you choose 2 books from a shelf of 40 books?
5) how many ways can 12 swimmers finish in first, second, and third place?
6) how many ways can mrs. sullivan choose two students from 27 to help put away calculators at the end of class?
7) how many different 12 - letter arrangements can be made using the letters in the word indianapolis?
8) how many different 11 - letter arrangements can be made using the letters in the word firecracker?
9) there are 18 offensive players on the hockey team. how many ways can the coach choose a left wing, center, and right wing to start the game?