Question

evaluate (_{12}c_4) and (_{10}p_4). (if necessary, consult a list of formulas.) (_{12}c_4 = square) (_{10}p_4 = square)

Explanation:

Step1: Recall combination formula

The formula for combinations is $_nC_r = \frac{n!}{r!(n-r)!}$. For $_{12}C_4$, $n=12$, $r=4$.
$$_{12}C_4 = \frac{12!}{4!(12-4)!} = \frac{12!}{4!8!}$$

Step2: Simplify combination calculation

Cancel $8!$ from numerator and denominator:
$$_{12}C_4 = \frac{12\times11\times10\times9}{4\times3\times2\times1} = \frac{11880}{24}$$

Step3: Recall permutation formula

The formula for permutations is $_nP_r = \frac{n!}{(n-r)!}$. For $_{10}P_4$, $n=10$, $r=4$.
$$_{10}P_4 = \frac{10!}{(10-4)!} = \frac{10!}{6!}$$

Step4: Simplify permutation calculation

Cancel $6!$ from numerator and denominator:
$$_{10}P_4 = 10\times9\times8\times7 = 5040$$

Answer:

$_{12}C_4 = 495$
$_{10}P_4 = 5040$