Question

evaluate the expression.
$_{15}mathrm{c}_4$

$_{15}mathrm{c}_4 = \square$ (simplify your answer.)

Explanation:

Step1: Recall Combination Formula

The formula for combinations is \( _nC_r=\frac{n!}{r!(n - r)!} \), where \( n = 15 \) and \( r = 4 \).

Step2: Substitute Values

Substitute \( n = 15 \) and \( r = 4 \) into the formula: \( _{15}C_4=\frac{15!}{4!(15 - 4)!}=\frac{15!}{4!×11!} \).

Step3: Simplify Factorials

Since \( n!=n\times(n - 1)\times\cdots\times1 \), we can simplify \( \frac{15!}{4!×11!}=\frac{15\times14\times13\times12\times11!}{4\times3\times2\times1\times11!} \). The \( 11! \) terms cancel out.

Step4: Calculate Numerator and Denominator

Calculate the numerator: \( 15\times14\times13\times12 = 32760 \). Calculate the denominator: \( 4\times3\times2\times1 = 24 \). Then divide: \( \frac{32760}{24}=1365 \).

Answer:

\( 1365 \)