Answer
$$252$$
Work Step by Step
$$\eqalign{
& C\left( {10,5} \right) \cr
& {\text{Use the combination formula }}C\left( {n,r} \right) = \frac{{n!}}{{\left( {n - r} \right)!r!}} \cr
& {\text{ }}C\left( {10,5} \right) = \frac{{10!}}{{\left( {10 - 5} \right)!5!}} \cr
& {\text{Evaluate and simplify}} \cr
& {\text{ }}C\left( {10,5} \right) = \frac{{10!}}{{5!5!}} \cr
& {\text{ }}C\left( {10,5} \right) = \frac{{5! \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10}}{{5!\left( {1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} \right)}} \cr
& {\text{ }}C\left( {10,5} \right) = \frac{{6 \cdot 7 \cdot 8 \cdot 9 \cdot 10}}{{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}} \cr
& {\text{ }}C\left( {10,5} \right) = 7 \cdot 2 \cdot 9 \cdot 2 \cr
& {\text{ }}C\left( {10,5} \right) = 252 \cr} $$
