$$ 1 \leftarrow \quad \begin{array}{l} ext { Evaluate the expression. } \ \begin{array}{l} ext { a) }{ }_{8} \mathrm{C}_{5} \ ext { b) }{ }_{8} \mathrm{P}_{5}\end{array}\end{array} $$

Question

$$ 1 \leftarrow \quad \begin{array}{l}	ext { Evaluate the expression. } \ \begin{array}{l}	ext { a) }{ }_{8} \mathrm{C}_{5} \ 	ext { b) }{ }_{8} \mathrm{P}_{5}\end{array}\end{array} $$

UpStudy AI · Accepted Answer

**Step 1: Evaluate $$ { }_{8}\mathrm{C}_{5} $$**

Recall that the combination formula is:
$$
{ }_{n}\mathrm{C}_{r} = \frac{n!}{r!(n-r)!}
$$
For $$ n = 8 $$ and $$ r = 5 $$:
$$
{ }_{8}\mathrm{C}_{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5! \cdot 3!}
$$

Calculate the factorials:
$$
8! = 40320,\quad 5! = 120,\quad 3! = 6
$$

Substitute the values:
$$
{ }_{8}\mathrm{C}_{5} = \frac{40320}{120 \cdot 6} = \frac{40320}{720} = 56
$$

---

**Step 2: Evaluate $$ { }_{8}\mathrm{P}_{5} $$**

Recall that the permutation formula is:
$$
{ }_{n}\mathrm{P}_{r} = \frac{n!}{(n-r)!}
$$
For $$ n = 8 $$ and $$ r = 5 $$:
$$
{ }_{8}\mathrm{P}_{5} = \frac{8!}{(8-5)!} = \frac{8!}{3!}
$$

Substitute the factorial values:
$$
{ }_{8}\mathrm{P}_{5} = \frac{40320}{6} = 6720
$$

---

**Final Answers:**

- a) $$ { }_{8}\mathrm{C}_{5} = 56 $$
- b) $$ { }_{8}\mathrm{P}_{5} = 6720 $$
a) $$ { }_{8}\mathrm{C}_{5} = 56 $$ b) $$ { }_{8}\mathrm{P}_{5} = 6720 $$

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