Given \[\left( 99 \right)5\]
\[99\] can be composed as the aggregate or distinction of two numbers then binomial hypothesis can be applied.
The given inquiry can be composed as \[99\text{ }=\text{ }100\text{ }-\text{ }1\]
\[\left( 99 \right)5\text{ }=\text{ }\left( 100\text{ }\text{ }1 \right)5\]
\[=\text{ }5C0\text{ }\left( 100 \right)5\text{ }\text{ }5C1\text{ }\left( 100 \right)4\text{ }\left( 1 \right)\text{ }+\text{ }5C2\text{ }\left( 100 \right)3\text{ }\left( 1 \right)2\text{ }\text{ }5C3\text{ }\left( 100 \right)2\text{ }\left( 1 \right)3\text{ }+\text{ }5C4\text{ }\left( 100 \right)\text{ }\left( 1 \right)4\text{ }\text{ }5C5\text{ }\left( 1 \right)5\]
\[=\text{ }\left( 100 \right)5\text{ }\text{ }5\text{ }\left( 100 \right)4\text{ }+\text{ }10\text{ }\left( 100 \right)3\text{ }\text{ }10\text{ }\left( 100 \right)2\text{ }+\text{ }5\text{ }\left( 100 \right)\text{ }\text{ }1\]
\[=\text{ }1000000000\text{ }\text{ }5000000000\text{ }+\text{ }10000000\text{ }\text{ }100000\text{ }+\text{ }500\text{ }\text{ }1\]
\[=\text{ }9509900499\]