Answer
$3,003$
Work Step by Step
Using $n!=n(n-1)(n-2)\cdot...(3)(2)(1),$ the given expression, $
\dfrac{15!}{10!\text{ }5!}
,$ simplifies to
\begin{align*}\require{cancel}
&
\dfrac{15(14)(13)(12)(11)(10!)}{10!\text{ }5(4)(3)(2)(1)}
\\\\&=
\dfrac{15(14)(13)(12)(11)(\cancel{10!})}{\cancel{10!}\text{ }5(4)(3)(2)(1)}
\\\\&=
\dfrac{15(14)(13)(\cancel{12})(11)}{5(\cancel4)(\cancel3)(2)(1)}
\\\\&=
\dfrac{\cancel{15}^3(14)(13)(11)}{\cancel5(2)(1)}
\\\\&=
\dfrac{3(\cancel{14}^7)(13)(11)}{(\cancel2)(1)}
\\\\&=
3(7)(13)(11)
\\&=
3003
.\end{align*}
Hence, the given expression evaluates to $
3,003
.$