Question

use the formula for $_{n}c_{r}$ to evaluate the given expression.
$_{10}c_{3}$
$_{10}c_{3}=\square$ (type an integer or a simplified fraction.)

Explanation:

Step1: Recall the combination formula

The formula for combination is \(_{n}C_{r}=\frac{n!}{r!(n - r)!}\). Here \(n = 10\) and \(r=3\). So \(_{10}C_{3}=\frac{10!}{3!(10 - 3)!}\).

Step2: Simplify the factorials

We know that \(n!=n\times(n - 1)\times\cdots\times1\). So \(10! = 10\times9\times8\times7!\), and \((10-3)!=7!\). Then \(_{10}C_{3}=\frac{10\times9\times8\times7!}{3!\times7!}\). Cancel out the \(7!\) terms. And \(3!=3\times2\times 1=6\).

Step3: Calculate the value

Now we have \(_{10}C_{3}=\frac{10\times9\times8}{6}\). \(10\times9\times8 = 720\), and \(\frac{720}{6}=120\).

Answer:

\(120\)