Question

evaluate the expression. (example 2)

1. \\( _8c_5 \\)

Explanation:

Step1: Recall Combination Formula

The combination formula is \( _nC_r=\frac{n!}{r!(n - r)!} \), where \( n!=n\times(n - 1)\times\cdots\times1 \) and \( 0!=1 \). For \( _8C_5 \), we have \( n = 8 \) and \( r = 5 \).

Step2: Substitute Values into Formula

Substitute \( n = 8 \) and \( r = 5 \) into the formula: \( _8C_5=\frac{8!}{5!(8 - 5)!}=\frac{8!}{5!3!} \).

Step3: Expand Factorials

Expand the factorials: \( 8! = 8\times7\times6\times5! \) and \( 3! = 3\times2\times1 \). So the expression becomes \( \frac{8\times7\times6\times5!}{5!\times3\times2\times1} \).

Step4: Simplify the Expression

Cancel out the \( 5! \) terms: \( \frac{8\times7\times6}{3\times2\times1} \). Calculate the numerator and denominator: \( 8\times7\times6 = 336 \) and \( 3\times2\times1 = 6 \). Then \( \frac{336}{6}=56 \).

Answer:

56