Permutation and Combination Worksheet Worksheet for 7th - 9th ... - Free Printable

Let’s solve each problem step by step. We’ll use the formulas for permutations and combinations:

- Permutation (order matters):
\( P(n, r) = \frac{n!}{(n - r)!} \)

- Combination (order does NOT matter):
\( C(n, r) = \frac{n!}{r!(n - r)!} \)

We’ll calculate each one carefully.

---

1. n = 3, r = 2

Permutation:
\( P(3, 2) = \frac{3!}{(3-2)!} = \frac{6}{1} = 6 \)

Combination:
\( C(3, 2) = \frac{3!}{2!(3-2)!} = \frac{6}{2 \cdot 1} = 3 \)

→ Permutation = 6, Combination = 3

---

2. n = 6, r = 4

Permutation:
\( P(6, 4) = \frac{6!}{(6-4)!} = \frac{720}{2} = 360 \)

Wait — let’s double-check that factorial:

6! = 720
(6-4)! = 2! = 2
So 720 ÷ 2 = 360 → Correct.

Combination:
\( C(6, 4) = \frac{6!}{4! \cdot 2!} = \frac{720}{24 \cdot 2} = \frac{720}{48} = 15 \)

Check: 6 choose 4 is same as 6 choose 2 → \( \frac{6×5}{2×1} = 15 \) → Correct.

→ Permutation = 360, Combination = 15

---

3. n = 3, r = 1

Permutation:
\( P(3, 1) = \frac{3!}{(3-1)!} = \frac{6}{2} = 3 \)

Actually, simpler: choosing 1 out of 3 in order → just 3 options.

Combination:
\( C(3, 1) = \frac{3!}{1! \cdot 2!} = \frac{6}{1 \cdot 2} = 3 \)

Also makes sense — picking 1 item from 3, no order → still 3 ways.

→ Permutation = 3, Combination = 3

---

4. n = 9, r = 2

Permutation:
\( P(9, 2) = \frac{9!}{(9-2)!} = \frac{9 × 8 × 7!}{7!} = 9 × 8 = 72 \)

Combination:
\( C(9, 2) = \frac{9!}{2! \cdot 7!} = \frac{9 × 8}{2 × 1} = 36 \)

→ Permutation = 72, Combination = 36

---

5. n = 5, r = 2

Permutation:
\( P(5, 2) = 5 × 4 = 20 \)

Combination:
\( C(5, 2) = \frac{5 × 4}{2 × 1} = 10 \)

→ Permutation = 20, Combination = 10

---

6. n = 7, r = 3

Permutation:
\( P(7, 3) = 7 × 6 × 5 = 210 \)

Combination:
\( C(7, 3) = \frac{7 × 6 × 5}{3 × 2 × 1} = \frac{210}{6} = 35 \)

→ Permutation = 210, Combination = 35

---

Final Answer:
For n=3, r=2: Permutation = 6, Combination = 3
For n=6, r=4: Permutation = 360, Combination = 15
For n=3, r=1: Permutation = 3, Combination = 3
For n=9, r=2: Permutation = 72, Combination = 36
For n=5, r=2: Permutation = 20, Combination = 10
For n=7, r=3: Permutation = 210, Combination = 35