Question

use the formula for $_{n}c_{r}$ to evaluate the given expression
$_{7}c_{3}$

$_{7}c_{3}=\square$ (type an integer or a simplified fraction.)

Explanation:

Step1: Recall the combination formula

The formula for combinations is \( _nC_r=\frac{n!}{r!(n - r)!} \), where \( n!=n\times(n - 1)\times\cdots\times1 \) and \( 0!=1 \). Here, \( n = 7 \) and \( r = 3 \).

Step2: Substitute values into the formula

Substitute \( n = 7 \) and \( r = 3 \) into the formula: \( _7C_3=\frac{7!}{3!(7 - 3)!}=\frac{7!}{3!4!} \).

Step3: Expand the factorials

Expand the factorials: \( 7! = 7\times6\times5\times4! \), \( 3! = 3\times2\times1 \), so \( \frac{7\times6\times5\times4!}{3\times2\times1\times4!} \).

Step4: Simplify the expression

Cancel out the \( 4! \) terms: \( \frac{7\times6\times5}{3\times2\times1} \). Calculate the numerator and denominator: numerator \( 7\times6\times5 = 210 \), denominator \( 3\times2\times1 = 6 \). Then \( \frac{210}{6}=35 \).

Answer:

\( 35 \)