Answer
The required solution is $210$
Work Step by Step
We know that the representation $_{n}{{C}_{r}}$ counts the number of assortments of n items taken r at a time.
And the number of assortment of n items taken r at a time can be evaluated as:
$_{n}{{C}_{r}}=\frac{n!}{\left( n-r \right)!r!}$
And the provided expression is $_{10}{{C}_{6}}$.
Here, $ n=10,r=6$.
Put the value of n, r in the above formula. Then:
$\begin{align}
& _{10}{{C}_{6}}=\frac{10!}{\left( 10-6 \right)!6!} \\
& =\frac{10!}{4!6!} \\
& =\frac{10\cdot 9\cdot 8\cdot 7\cdot 6!}{4\cdot 3\cdot 2\cdot 1\cdot 6!}
\end{align}$
And simplify further, $\begin{align}
& \frac{10\cdot 9\cdot 8\cdot 7\cdot 6!}{4\cdot 3\cdot 2\cdot 1\cdot 6!}=\frac{10\cdot 9\cdot 8\cdot 7}{4\cdot 3\cdot 2\cdot 1} \\
& =\frac{5040}{24} \\
& =210
\end{align}$
Hence, $_{10}{{C}_{6}}=210$
