Question

evaluate each expression. sample problem ( _8p_3 ) ( \frac{8!}{(8-3)!} = \frac{8!}{5!} = 336 ) ( _{11}c_4 ) enter the answer in the space provided. use numbers instead of words.

Explanation:

Step1: Recall the combination formula

The formula for combinations is \(_{n}C_{r}=\frac{n!}{r!(n - r)!}\), where \(n = 11\) and \(r=4\).

Step2: Substitute the values into the formula

Substitute \(n = 11\) and \(r = 4\) into the formula: \(_{11}C_{4}=\frac{11!}{4!(11 - 4)!}=\frac{11!}{4!×7!}\)

Step3: Expand the factorials

We know that \(n!=n\times(n - 1)\times\cdots\times1\), so \(11! = 11\times10\times9\times8\times7!\) and \(4! = 4\times3\times2\times1\). Substituting these into the formula gives \(\frac{11\times10\times9\times8\times7!}{4\times3\times2\times1\times7!}\)

Step4: Cancel out the common terms

The \(7!\) terms in the numerator and denominator cancel out, leaving \(\frac{11\times10\times9\times8}{4\times3\times2\times1}\)

Step5: Calculate the numerator and denominator

Calculate the numerator: \(11\times10\times9\times8=7920\)
Calculate the denominator: \(4\times3\times2\times1 = 24\)

Step6: Divide the numerator by the denominator

\(\frac{7920}{24}=330\)

Answer:

330