Question

evaluate the expression.
13c3
13c3 = (simplify your answer.)

Explanation:

Step1: Recall the combination formula

The combination formula is \(_{n}C_{r}=\frac{n!}{r!(n - r)!}\), where \(n = 13\) and \(r=3\).
So, \(_{13}C_{3}=\frac{13!}{3!(13 - 3)!}\).

Step2: Simplify the factorials

First, \(13-3 = 10\), so \(_{13}C_{3}=\frac{13!}{3!×10!}\).
Since \(n!=n\times(n - 1)\times\cdots\times1\), then \(13! = 13\times12\times11\times10!\).
Substitute \(13!\) into the formula: \(_{13}C_{3}=\frac{13\times12\times11\times10!}{3!×10!}\).
Cancel out the \(10!\) terms: \(_{13}C_{3}=\frac{13\times12\times11}{3!}\).
And \(3!=3\times2\times1 = 6\).

Step3: Calculate the value

Now, \(_{13}C_{3}=\frac{13\times12\times11}{6}\).
\(13\times12\times11=1716\), and \(\frac{1716}{6}=286\).

Answer:

\(286\)