Permutations and Combinations Notes and Worksheets - Lindsay Bowden - Free Printable

Let’s solve each problem step by step.

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Problem 1: How many ways can the letters in the word MAINE be arranged if you use all 5 letters (and repetition is allowed)?

Wait — this says “repetition is allowed”, but the word MAINE has 5 *different* letters: M, A, I, N, E. If repetition is allowed, that means we can reuse any letter as many times as we want when making arrangements of 5 letters.

But note: The question says “use all 5 letters” — which might mean we are arranging exactly 5 positions, and since repetition is allowed, each position can be any of the 5 letters.

So:

- n = number of things to choose from = 5 letters (M, A, I, N, E)
- r = how many things you choose = 5 positions
- Repetition allowed → formula: n^r

So:
5⁵ = 5 × 5 × 5 × 5 × 5 = 3125

✔ But wait — let’s double-check the wording: “if you use all 5 letters”. That could be misinterpreted. In permutations with repetition allowed, “using all 5 letters” doesn’t mean each letter must appear once — it just means we’re filling 5 slots, and we can repeat. Since the problem explicitly says “repetition is allowed”, we go with n^r.

Answer for #1: 3125

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Problem 2: How many different 3-digit codes can you make with the single digit whole numbers (without repeating)?

Single digit whole numbers = 0 through 9 → that’s 10 digits.

We want 3-digit codes, no repetition.

Important: Can a code start with 0? Yes — because it’s a “code”, not necessarily a number. So 012 is a valid 3-digit code.

So:

- n = 10 (digits 0–9)
- r = 3 (positions in the code)
- No repetition → formula: n! / (n - r)!

So:
₁₀P₃ = 10! / (10 - 3)! = 10! / 7! = 10 × 9 × 8 = 720

✔ Check: First digit: 10 choices, second: 9 left, third: 8 left → 10×9×8=720

Answer for #2: 720

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Problem 3: How many ways can 3 people win 1st, 2nd, and 3rd place?

This implies there are more than 3 people total? Wait — actually, the problem doesn’t specify how many people are competing. But looking back at the example above: “There are 8 people running a race...” — so maybe this is continuing that context?

Wait — Problem 3 stands alone: “How many ways can 3 people win 1st, 2nd, and 3rd place?”

That phrasing is ambiguous. Does it mean:

A) There are only 3 people, and they are being assigned to 1st, 2nd, 3rd? → Then it’s 3! = 6

OR

B) There are more people (like 8 from previous example), and we’re choosing 3 winners out of them?

Looking at the worksheet structure — Problem 3 comes after Problem 2, and before Problem 4 (“8 students sit in 4 seats”). Also, the example above the table used 8 people for 1st/2nd/3rd.

But Problem 3 doesn’t mention a total number. Hmm.

Wait — perhaps it’s implying: “If there are 3 people, how many ways can they be ordered into 1st, 2nd, 3rd?” That would be 3!.

Alternatively, maybe it’s a trick — if only 3 people are running, then yes, 3! = 6.

But let’s look at the pattern:

- Example in table: 8 people → 1st, 2nd, 3rd → uses permutation without repetition: ₈P₃

- Problem 4: 8 students in 4 seats → ₈P₄

So likely, Problem 3 is meant to be similar — but it doesn’t say how many total people.

Wait — re-read: “How many ways can 3 people win 1st, 2nd, and 3rd place?”

This probably means: There are 3 distinct people, and we assign them to 3 distinct places → order matters → 3! = 6

Because if it were selecting 3 winners from a larger group, it would say “from X people”.

Also, in the absence of a specified total, the most logical interpretation is that we have exactly 3 people to assign to 3 positions.

So:

n = 3, r = 3, no repetition → 3! = 6

✔ Answer for #3: 6

*(Note: If the problem intended a larger pool, it should have specified. Given the wording, 6 is correct.)*

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Problem 4: How many ways can 8 students sit in 4 seats?

This is classic permutation without repetition.

- n = 8 students
- r = 4 seats
- Order matters (who sits where)
- No repetition (one student per seat)

Formula: n! / (n - r)! = ₈P₄

Calculate:

₈P₄ = 8 × 7 × 6 × 5 = ?

8 × 7 = 56
56 × 6 = 336
336 × 5 = 1680

✔ Answer for #4: 1680

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Now, let’s compile final answers.

Final Answer:
1. 3125
2. 720
3. 6
4. 1680