Question

1. evaluate (^{10}c_5). (^{10}c_5 =) enter your next step here

Explanation:

Step1: Recall the combination formula

The formula for combinations is \( _nC_r=\frac{n!}{r!(n - r)!} \). For \( _{10}C_5 \), we have \( n = 10 \) and \( r=5 \).

Step2: Substitute the values into the formula

Substitute \( n = 10 \) and \( r = 5 \) into the formula: \( _{10}C_5=\frac{10!}{5!(10 - 5)!}=\frac{10!}{5!5!} \)

Step3: Expand the factorials

We know that \( n!=n\times(n - 1)\times\cdots\times1 \). So, \( 10! = 10\times9\times8\times7\times6\times5! \) and \( 5! = 5\times4\times3\times2\times1 \). Substituting these into the formula:
\( _{10}C_5=\frac{10\times9\times8\times7\times6\times5!}{5!\times5\times4\times3\times2\times1} \)

Step4: Cancel out the common terms

The \( 5! \) terms in the numerator and denominator cancel out. Then we have:
\( _{10}C_5=\frac{10\times9\times8\times7\times6}{5\times4\times3\times2\times1} \)

Step5: Calculate the numerator and denominator

Calculate the numerator: \( 10\times9\times8\times7\times6=30240 \)
Calculate the denominator: \( 5\times4\times3\times2\times1 = 120 \)
Then, \( \frac{30240}{120}=252 \)

Answer:

\( 252 \)