Question

use the formula for $_{n}c_{r}$ to evaluate the given expression.
$_{20}c_{4}$

$_{20}c_{4}=\square$ (type an integer or a simplified fraction.)

Explanation:

Step1: Write the combination formula

The formula for combination is \(_{n}C_{r}=\frac{n!}{r!(n - r)!}\). Here \(n = 20\) and \(r=4\), so \(_{20}C_{4}=\frac{20!}{4!(20 - 4)!}\).

Step2: Simplify the factorials

Since \(20! = 20\times19\times18\times17\times16!\) and \((20 - 4)!=16!\), then \(_{20}C_{4}=\frac{20\times19\times18\times17\times16!}{4!\times16!}\). Cancel out the \(16!\) terms. And \(4!=4\times3\times2\times1 = 24\).

Step3: Calculate the value

\(_{20}C_{4}=\frac{20\times19\times18\times17}{24}\).
\(20\times19\times18\times17=(20\times19)\times(18\times17)=380\times306 = 116280\).
\(\frac{116280}{24}=4845\).

Answer:

\(4845\)