Question

If we have \[{}^{n}{{C}_{12}}={}^{n}{{C}_{8}}\], then $n$ is equal to
A. 12
B. 26
C. 6
D. 20

Answer 1

Hint: We first discuss the general form of combination and its general meaning with the help of variables. We express the mathematical notion with respect to the factorial form of ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\times \left( n-r \right)!}$. We try to find the value of ${}^{n}{{C}_{n-r}}$. We complete the multiplication and find the relation.

Complete step-by-step solution:
We first try to find the general form of combination and its general meaning and then we put the values to find the solution.
The general form of combination is ${}^{n}{{C}_{r}}$. It’s used to express the notion of choosing r objects out of n objects. The value of ${}^{n}{{C}_{r}}$ expresses the number of ways the combination of those objects can be done.
The simplified form of the mathematical expression ${}^{n}{{C}_{r}}$ is ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\times \left( n-r \right)!}$.
Here the term $n!$ defines the notion of multiplication of first n natural numbers.
This means $n!=1\times 2\times 3\times ....\times n$.
The arrangement of those chosen objects is not considered in case of combination. That part is involved in permutation.
We now try to find the value of ${}^{n}{{C}_{n-r}}$ which is ${}^{n}{{C}_{n-r}}=\dfrac{n!}{\left( n-r \right)!\times \left[ n-\left( n-r \right) \right]!}=\dfrac{n!}{\left( n-r \right)!\times r!}$.
Therefore, ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\times \left( n-r \right)!}={}^{n}{{C}_{n-r}}$.
For \[{}^{n}{{C}_{12}}={}^{n}{{C}_{8}}\], we can say $n=12+8=20$. The correct option is D.

Note: There are some constraints in the form of ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\times \left( n-r \right)!}$. The general conditions are $n\ge r\ge 0;n\ne 0$. Also, we need to remember the fact that the notion of choosing r objects out of n objects is exactly equal to the notion of choosing $\left( n-r \right)$ objects out of n objects. The mathematical expression is ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\times \left( n-r \right)!}={}^{n}{{C}_{n-r}}$.