$$ \quad \frac{{ }_{8} \mathrm{C}_{5}}{{ }_{18} \mathrm{C}_{5}}=\square $$ (Simplify your answer.)

Question

UpStudy AI · Accepted Answer

We start with the combination formulas:

$$
{8 \choose 5}=\frac{8!}{5! \cdot 3!}\quad \text{and}\quad {18 \choose 5}=\frac{18!}{5! \cdot 13!}
$$

Thus, the ratio becomes

$$
\frac{{8 \choose 5}}{{18 \choose 5}}=\frac{\frac{8!}{5! \cdot 3!}}{\frac{18!}{5! \cdot 13!}}=\frac{8!}{5! \cdot 3!}\cdot\frac{5! \cdot 13!}{18!}=\frac{8! \cdot 13!}{3! \cdot 18!}
$$

Notice that we can simplify by calculating each combination. First, note that

$$
{8 \choose 5}={8 \choose 3}=\frac{8 \cdot 7 \cdot 6}{3!}=\frac{8 \cdot 7 \cdot 6}{6}=56
$$

Next, compute $${18 \choose 5}$$:

$$
{18 \choose 5}=\frac{18 \cdot 17 \cdot 16 \cdot 15 \cdot 14}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}=\frac{18 \cdot 17 \cdot 16 \cdot 15 \cdot 14}{120}
$$

Calculating the numerator step by step:

$$
18 \cdot 17=306,\quad 306 \cdot 16=4896,\quad 4896 \cdot 15=73440,\quad 73440 \cdot 14=1028160
$$

Thus,

$$
{18 \choose 5}=\frac{1028160}{120}=8568
$$

Now, the ratio becomes

$$
\frac{{8 \choose 5}}{{18 \choose 5}}=\frac{56}{8568}
$$

Simplify by dividing numerator and denominator by 56:

$$
\frac{56}{8568}=\frac{1}{\frac{8568}{56}}=\frac{1}{153}
$$

Thus, the simplified answer is

$$
\frac{1}{153}
$$
$$ \frac{1}{153} $$

(Simplify your answer.)

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