How do you find the binomial coefficient of \[\left( {\begin{array}{*{20}{c}}
  {10} \\
  6
\end{array}} \right)\]?

Hint: In this question, we are asked to find binomial coefficient. To further solve the question we need to know what binomial coefficient means. Binomial coefficient\[\left( {\begin{array}{*{20}{c}}
  n \\
  k
\end{array}} \right)\] defines the number of ways in which we can pick $k$ unordered outcomes from $n$ number of possibilities. For example- to choose a committee of $2$ people from total of $4$ number, if can be written as\[\left( {\begin{array}{*{20}{c}}
  4 \\
  2
\end{array}} \right)\]. Binomial Theorem is a way of expanding a Binomial expression Binomial coefficient is used when power$4$ of an expression becomes too large to be calculated manually. Using the formula the expression can be raised to any finite power. To find a binomial coefficient we can use Pascal’s Triangle .

Formula used: \[\left( {\begin{array}{*{20}{c}}
  n \\
  k
\end{array}} \right) = \dfrac{{n!}}{{k!(n - k)!}}\]

Complete step by step solution:
We are given,
\[\left( {\begin{array}{*{20}{c}}
  {10} \\
  6
\end{array}} \right) = \] \[\left( {\begin{array}{*{20}{c}}
  {10} \\
  6
\end{array}} \right)\]
Where,
$
  n = 10 \\
  k = 6 \\
 $
We’ll put the values in the formula
\[\left( {\begin{array}{*{20}{c}}
  n \\
  k
\end{array}} \right) = \dfrac{{n!}}{{k!(n - k)!}}\]
\[ \Rightarrow \left( {\begin{array}{*{20}{c}}
  {10} \\
  6
\end{array}} \right) = \dfrac{{10!}}{{6!(10 - 6)!}}\]
\[ \Rightarrow \left( {\begin{array}{*{20}{c}}
  {10} \\
  6
\end{array}} \right) = \dfrac{{10!}}{{6!(4)!}}\]
\[ \Rightarrow \left( {\begin{array}{*{20}{c}}
  {10} \\
  6
\end{array}} \right) = \dfrac{{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{6 \times 5 \times 4 \times 3 \times 2 \times 1(4 \times 3 \times 2 \times 1)}}\]
After, cancelling out the same terms from the numerator and denominator
\[ \Rightarrow \left( {\begin{array}{*{20}{c}}
  {10} \\
  6
\end{array}} \right) = \dfrac{{10 \times 9 \times 8 \times 7}}{{(4 \times 3 \times 2 \times 1)}}\]
\[ \Rightarrow \left( {\begin{array}{*{20}{c}}
  {10} \\
  6
\end{array}} \right) = 210\]

Note: Some properties of binomial expansion are-
Total number of terms in the expansion of expression is one more than the power or exponent of expression.
Sum of the exponents or power in each term of the expression adds up to n.
The first and last term of expanded expression has one as coefficient.
Factorial of a number is the product of all the numbers from 1 to that number. It is represented by the symbol “!” symbol. Factorial can only be calculated for non-negative numbers and factorial of zero is one. Also, dividing or multiplying the numerator and denominator of a fraction by the same number does not change the value of the fraction.