Question

evaluate the expression.
\\(\frac{_7c_2 \cdot _3c_1}{_9c_3}\\)
\\(\frac{_7c_2 \cdot _3c_1}{_9c_3} = \square\\) (type an integer or a simplified fraction.)

Explanation:

Step1: Calculate $_7C_2$

Combination formula: $_nC_k = \frac{n!}{k!(n-k)!}$.
$_7C_2 = \frac{7!}{2!5!} = \frac{7×6}{2×1} = 21$.

Step2: Calculate $_3C_1$

$_3C_1 = \frac{3!}{1!2!} = \frac{3}{1} = 3$.

Step3: Calculate $_9C_3$

$_9C_3 = \frac{9!}{3!6!} = \frac{9×8×7}{3×2×1} = 84$.

Step4: Compute the expression

$\frac{21×3}{84} = \frac{63}{84} = \frac{3}{4}$.

Answer:

$\frac{3}{4}$