How do you evaluate \[{}^{12}{C_6}\]?

Hint: The given question describes the operation of addition/ subtraction/ multiplication/ division, substituting the mathematical formula and factorial calculation. To solve this question we need to know the factorial calculations and exceptions in factorial operation. Also, we should try to cancel the same term in the numerator and denominator.

Complete answer:
The given question is given below,
\[{}^{12}{C_6} = ?\]\[ \to \left( 1 \right)\]
The above-mentioned question is in the form of \[{}^n{C_r}\]
We know that,
\[{}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}} \to \left( 2 \right)\]
Let’s compare the\[\left( 1 \right)and\left( 2 \right)\], we get
\[\left( 1 \right) \to {}^{12}{C_6} = ?\]
\[\left( 2 \right) \to {}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}\]
By comparing these two-equation we find the values of r and n.
So, the values of \[r = 6\] and \[n = 12\]
Let’s substitute the values of r and n in equation (2), we get
\[
  {}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}} \\
  {}^{12}{C_6} = \dfrac{{12!}}{{6!(12 - 6)!}} = \dfrac{{12!}}{{6! \times 6!}} \to \left( 3 \right) \\
 \]
Here, we have to find the value of 12! In the numerator part which is given below,
\[12! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10 \times 11 \times 12\]
Next, we have to find the value of 6! In the denominator part which is given below,
\[6! = 1 \times 2 \times 3 \times 4 \times 5 \times 6\]
So, the value of 12! And 6! Are substitute in equation (3), we get
\[{}^{12}{C_6} = \dfrac{{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10 \times 11 \times 12}}{{\left( {1 \times 2 \times 3 \times 4 \times 5 \times 6} \right)\left( {1 \times 2 \times 3 \times 4 \times 5 \times 6} \right)}}\]
Here, the term \[\left( {1 \times 2 \times 3 \times 4 \times 5 \times 6} \right)\] is common in the numerator and the denominator. So, the term can be cancelled with each other. So, we get
\[{}^{12}{C_6} = \dfrac{{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10 \times 11 \times 12}}{{\left( {1 \times 2 \times 3 \times 4 \times 5 \times 6} \right)\left( {1 \times 2 \times 3 \times 4 \times 5 \times 6} \right)}}\]
\[{}^{12}{C_6} = \dfrac{{7 \times 8 \times 9 \times 10 \times 11 \times 12}}{{\left( {1 \times 2 \times 3 \times 4 \times 5 \times 6} \right)}}\]
So, we get
\[{}^{12}{C_6} = \dfrac{{665280}}{{720}} = 924\]
So, the final answer is \[{}^{12}{C_6} = 924\]

Note: To solve this given question we would use the operation of addition/ subtraction/ multiplication/ division. Also, take care while doing the factorial calculation. The value of zero factorial is 1. If the same term is present in the numerator and denominator we can easily cancel the term with the replacement of “1”. Note that the denominator would not be equal to zero.