Math worksheet focusing on combinations and permutations, featuring exercises to calculate the number of possible unordered combinations and ordered permutations.

Let's solve the problems step by step.

---

Part 1: Combinations (Unordered)

Combinations are used when the order does not matter. The formula for combinations is:

$$
\binom{n}{r} = \frac{n!}{r!(n - r)!}
$$

Where:
- $ n $ = total number of items
- $ r $ = number of items to choose
- $ ! $ denotes factorial (e.g., $ 5! = 5 \times 4 \times 3 \times 2 \times 1 $)

---

#### 1.) $ \binom{6}{3} $

$$
\binom{6}{3} = \frac{6!}{3!(6 - 3)!} = \frac{6!}{3! \cdot 3!}
= \frac{720}{6 \cdot 6} = \frac{720}{36} = 20
$$

✔ Answer: 20

---

#### 2.) $ \binom{4}{2} $

$$
\binom{4}{2} = \frac{4!}{2!(4 - 2)!} = \frac{24}{2 \cdot 2} = \frac{24}{4} = 6
$$

✔ Answer: 6

---

#### 3.) $ \binom{8}{3} $

$$
\binom{8}{3} = \frac{8!}{3!(8 - 3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56
$$

✔ Answer: 56

---

#### 4.) $ \binom{2}{1} $

$$
\binom{2}{1} = \frac{2!}{1!(2 - 1)!} = \frac{2}{1 \cdot 1} = 2
$$

✔ Answer: 2

---

#### 5.) $ \binom{7}{4} $

$$
\binom{7}{4} = \frac{7!}{4!(7 - 4)!} = \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} = \frac{840}{24} = 35
$$

✔ Answer: 35

---

✔ Combination Answers:

1. 20
2. 6
3. 56
4. 2
5. 35

---

Part 2: Permutations (Ordered)

Permutations are used when the order does matter. The formula is:

$$
P(n, r) = \frac{n!}{(n - r)!}
$$

---

#### 1.) $ P(5, 3) $

$$
P(5, 3) = \frac{5!}{(5 - 3)!} = \frac{120}{2!} = \frac{120}{2} = 60
$$

✔ Answer: 60

---

#### 2.) $ P(3, 2) $

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

✔ Answer: 6

---

#### 3.) $ P(7, 3) $

$$
P(7, 3) = \frac{7!}{(7 - 3)!} = \frac{7 \times 6 \times 5}{1} = 210
$$

✔ Answer: 210

---

#### 4.) $ P(9, 1) $

$$
P(9, 1) = \frac{9!}{(9 - 1)!} = \frac{9!}{8!} = 9
$$

✔ Answer: 9

---

#### 5.) $ P(8, 2) $

$$
P(8, 2) = \frac{8!}{(8 - 2)!} = \frac{8 \times 7}{1} = 56
$$

✔ Answer: 56

---

✔ Permutation Answers:

1. 60
2. 6
3. 210
4. 9
5. 56

---

✔ Final Answers:

#### Combinations:
1. $ \binom{6}{3} = 20 $
2. $ \binom{4}{2} = 6 $
3. $ \binom{8}{3} = 56 $
4. $ \binom{2}{1} = 2 $
5. $ \binom{7}{4} = 35 $

#### Permutations:
1. $ P(5, 3) = 60 $
2. $ P(3, 2) = 6 $
3. $ P(7, 3) = 210 $
4. $ P(9, 1) = 9 $
5. $ P(8, 2) = 56 $

Let me know if you'd like a visual explanation or help with understanding factorials!