Algebra 4 Permutations & Combinations Worksheet featuring ten word problems on permutations and combinations.

Let’s go through each problem one by one. Since the user didn’t specify which problem to solve, and all 10 are listed, I’ll assume they want all problems solved — but since the instruction says “solve the problem” (singular), and no specific one is highlighted, I’ll pick Problem #1 as a representative example to solve fully, following the rules.

But wait — looking again at the user’s message: it says “You are an educational assistant helping a student solve homework problems.” and then “Solve the problem accurately.” — plural or singular? The image has 10 problems. But the response format rules say: if ExplainAnswer is false, omit that section. And GeneratePractice is also false. So we’re only supposed to give the Final Answer — but only after working through reasoning.

Actually, re-reading the critical rule:
> CRITICAL: First work through the problem step by step, check your calculations carefully, and ONLY THEN provide the final answer.

And:
> If you're solving math or logic problems, verify your answer before writing it in the Final Answer section.

So even though ExplainAnswer is false, I still need to do the reasoning internally — but not write it out? Wait, no — the instruction says: “Complete ALL reasoning and explanations BEFORE writing the Final Answer.” But then it says: “If ExplainAnswer is true: Explanation: ...” — implying that if it’s false, we don’t write the explanation.

This is ambiguous. But let’s look at the structure:

The response format says:
→ Always follow the structure below.
→ If an option is false, completely omit that section.

So if ExplainAnswer is false, we skip the Explanation section. Same for TeachTopic and GeneratePractice.

Then it says:
Final Answer:
[Provide only the final correct answer — this should come AFTER all explanations and reasoning.]

But if we omit the explanation section because ExplainAnswer is false, how can we have reasoning before the final answer?

I think the intent is: even if you don’t output the explanation, you still must do the reasoning internally, and then output only the Final Answer — but the Final Answer should be based on correct reasoning.

However, the user might expect all 10 problems solved? But that would be 10 final answers. The format doesn’t specify.

Looking back: the user uploaded an image with 10 problems. The task is to “solve the problem” — likely meaning solve each one. But the response format only allows one Final Answer section.

This is confusing. Perhaps I should solve all 10 and list them? But the format says “Final Answer: [only the final correct answer]” — singular.

Alternatively, maybe the user wants me to solve one problem as an example? But which one?

Wait — perhaps I misread. The user says: “You are an educational assistant helping a student solve homework problems.” — plural. And the image has 10 problems. So likely, I need to solve all 10.

But the response format only has one Final Answer section. That won’t work.

Perhaps I should interpret “the problem” as the entire worksheet, and provide answers to all 10.

But to comply with the format, I’ll solve all 10 problems step by step (internally), and then in the Final Answer section, list all 10 answers clearly.

Let me do that.

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Problem 1:
Fifteen students ask to visit the admissions representative from WSU. Each visit includes one student. In how many ways can ten time slots be assigned?

This is permutation: we are assigning 10 distinct time slots to 10 students chosen from 15, and order matters (since time slots are distinct).

So, P(15,10) = 15! / (15-10)! = 15! / 5!

Calculate:
15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 = ?

Let me compute step by step:

15 × 14 = 210
210 × 13 = 2730
2730 × 12 = 32760
32760 × 11 = 360360
360360 × 10 = 3,603,600
3,603,600 × 9 = 32,432,400
32,432,400 × 8 = 259,459,200
259,459,200 × 7 = 1,816,214,400
1,816,214,400 × 6 = 10,897,286,400

So, 10,897,286,400

But let me verify with calculator if possible — since I’m AI, I can compute exactly.

P(15,10) = 15×14×13×12×11×10×9×8×7×6

We can group:

(15×14×13×12×11) = 360360
(10×9×8×7×6) = 30240
Then 360360 × 30240

But better to compute sequentially:

Start over:

15 × 14 = 210
210 × 13 = 2730
2730 × 12 = 32760
32760 × 11 = 360360
360360 × 10 = 3,603,600
3,603,600 × 9 = 32,432,400
32,432,400 × 8 = 259,459,200
259,459,200 × 7 = 1,816,214,400
1,816,214,400 × 6 = 10,897,286,400

Yes.

So answer for #1: 10,897,286,400

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Problem 2:
The prom committee has four sites available for the banquet and three sites for the dance. How many arrangements are possible for the banquet and dance?

Assuming they choose one site for banquet and one for dance, and sites are distinct.

So, 4 choices for banquet, 3 for dance. Total arrangements: 4 × 3 = 12

Answer: 12

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Problem 3:
How many different teams of 11 players can be chosen from a soccer squad of 16? (It doesn't matter when you are chosen- all that matters if you are on the team or not)

This is combination: C(16,11)

C(16,11) = C(16,5) because C(n,k)=C(n,n-k)

C(16,5) = 16! / (5! * 11!) = (16×15×14×13×12)/(5×4×3×2×1)

Compute numerator: 16×15=240, ×14=3360, ×13=43680, ×12=524160

Denominator: 120

524160 / 120 = 4368

So, 4368

Answer: 4368

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Problem 4:
Suppose you find seven articles related to the topic of your research paper. In how many ways can you choose five articles to read?

Combination: C(7,5) = C(7,2) = (7×6)/2 = 21

Answer: 21

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Problem 5:
For a band camp, you can choose two or three roommates from a group of 25 friends. In how many ways can you choose?

So, choose 2 OR choose 3.

C(25,2) + C(25,3)

C(25,2) = (25×24)/2 = 300

C(25,3) = (25×24×23)/(3×2×1) = (25×24×23)/6

24/6=4, so 25×4×23 = 100×23 = 2300

Total: 300 + 2300 = 2600

Answer: 2600

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Problem 6:
A salad bar offers eight choices of toppings for lettuce. In how many ways can you choose four or five toppings? (The salad toppings get mixed together later- so order does not matter.)

So, C(8,4) + C(8,5)

C(8,4) = 70 (since 8×7×6×5/(4×3×2×1) = 70)

C(8,5) = C(8,3) = (8×7×6)/(3×2×1) = 56

Total: 70 + 56 = 126

Answer: 126

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Problem 7:
In how many ways can four distinct positions for a relay race be assigned from a team of nine runners?

Permutation: P(9,4) = 9×8×7×6 = 3024

Answer: 3024

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Problem 8:
A committee must choose 3 finalists from 15 scholarship candidates. How many ways can the committee choose the three finalists?

Combination: C(15,3) = (15×14×13)/(3×2×1) = (15×14×13)/6

15/3=5, so 5×14×13 / 2 = 5×7×13 = 35×13 = 455

Answer: 455

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Problem 9:
In how many ways can 6 students be seated at one side of a table with 4 chairs?

First, choose 4 students out of 6: C(6,4)

Then, arrange them in 4 chairs: 4!

So, C(6,4) × 4! = [C(6,2)] × 24 = 15 × 24 = 360

C(6,4) = 15, yes.

Alternatively, P(6,4) = 6×5×4×3 = 360

Same thing.

Answer: 360

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Problem 10:
A traveler can choose from three airlines, five hotels, and four rental car companies. How many arrangements of these services are possible?

Assuming one choice from each category.

So, 3 × 5 × 4 = 60

Answer: 60

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Now, compiling all answers:

1. 10,897,286,400
2. 12
3. 4,368
4. 21
5. 2,600
6. 126
7. 3,024
8. 455
9. 360
10. 60

Since the response format requires a Final Answer section, and I’ve done the reasoning, I’ll list them clearly.

But the format says “Final Answer: [only the final correct answer]” — perhaps I should present them as a list.

Given that, I’ll write:

Final Answer:
1. 10897286400
2. 12
3. 4368
4. 21
5. 2600
6. 126
7. 3024
8. 455
9. 360
10. 60

To make it neat, I'll write without commas for consistency, as sometimes large numbers are written without commas in math contexts.

But 10,897,286,400 is fine with commas for readability, but in final answer, perhaps without.

Looking at the problem, it's a worksheet, so likely expects numerical answers.

I think it's acceptable to write with commas for large numbers, but to be safe, I'll write without.

Actually, in the calculation, I had 10,897,286,400 — which is 10897286400.

Similarly, others are small.

So:

Final Answer:
1. 10897286400
2. 12
3. 4368
4. 21
5. 2600
6. 126
7. 3024
8. 455
9. 360
10. 60

But to match the format, perhaps just list them in order.

Since the user might expect only one answer, but I think solving all is correct.

Another thought: perhaps the user wants me to solve a specific problem, but none is specified. In that case, I should solve the first one as an example.

But the instruction says "helping a student solve homework problems" — plural.

I think providing all is better.

To comply strictly, I'll provide the answers for all 10.

So, final output: