Question

evaluate the factorial expression.
\\(\frac{15!}{2!13!}\\)
\\(\frac{15!}{2!13!} = \square\\) (simplify your answer.)

Explanation:

Step1: Recall factorial definition

The factorial of a non - negative integer \(n\), denoted as \(n!\), is defined as \(n!=n\times(n - 1)\times(n - 2)\times\cdots\times1\). So, we can write \(15!=15\times14\times13!\).

Step2: Substitute into the expression

Substitute \(15!=15\times14\times13!\) into the given expression \(\frac{15!}{2!13!}\). We get \(\frac{15\times14\times13!}{2!×13!}\).

Step3: Cancel out common terms

The \(13!\) terms in the numerator and denominator cancel out, leaving us with \(\frac{15\times14}{2!}\).

Step4: Evaluate \(2!\)

We know that \(2!=2\times1 = 2\). So the expression becomes \(\frac{15\times14}{2}\).

Step5: Calculate the result

First, calculate \(15\times14=210\). Then, divide by 2: \(\frac{210}{2}=105\).

Answer:

\(105\)