12. If $$ \frac{{ }^{9} \mathrm{C}_{4}}{{ }^{8} \mathrm{C}_{3}}=4(\mathrm{k}) $$, then $$ \mathrm{k}= $$

Question

UpStudy AI · Accepted Answer

We start with the given equation:

$$
\frac{{}^{9}C_{4}}{{}^{8}C_{3}} = 4k
$$

**Step 1: Compute $${}^{9}C_{4}$$.**

$$
{}^{9}C_4 = \frac{9!}{4!(9-4)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = \frac{3024}{24} = 126
$$

**Step 2: Compute $${}^{8}C_{3}$$.**

$$
{}^{8}C_3 = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56
$$

**Step 3: Substitute the computed values into the equation.**

$$
\frac{126}{56} = 4k
$$

Simplify the fraction by dividing numerator and denominator by 14:

$$
\frac{126 \div 14}{56 \div 14} = \frac{9}{4} = 4k
$$

**Step 4: Solve for $$ k $$.**

$$
4k = \frac{9}{4} \quad \Longrightarrow \quad k = \frac{9}{4 \times 4} = \frac{9}{16}
$$

Thus,

$$
k = \frac{9}{16}
$$
$$ k = \frac{9}{16} $$

If , then