Question

How do you find the binomial coefficient of \[^{12}{{C}_{5}}\] ?

Answer 1

Hint: In the above question we have to find the binomial coefficient of \[^{12}{{C}_{5}}\] . To find the binomial coefficient for \[{{(a+b)}^{n}}\], we will use the nth row of the pascal’s triangle. Here we can also use the formula of permutations and combinations to get the coefficient.

Complete step by step answer:
The question is from the concept of binomial theorem. Before solving the question let us get familiar with the binomial expansion. We know that binomial is a polynomial which has only two terms in it and binomial expansion is expanding the binomial expression using binomial theorem.
Here in order to find the coefficient of \[^{12}{{C}_{5}}\] we will use the formula
\[^{n}{{C}_{m}}=\dfrac{n!}{(n-m)!m!}\].
Here we have to find the factorial so factorial is basically the product of all the positive integers less than or equal to a given positive integer. It is represented by an exclamation mark “\[!\]”. Practical application of factorial is in permutations and combinations.
So, the coefficient is
\[\begin{align}
  & ^{n}{{C}_{m}}=\dfrac{n!}{(n-m)!m!} \\
 & {{\Rightarrow }^{12}}{{C}_{5}}=\dfrac{12!}{(12-5)!5!} \\
 & {{\Rightarrow }^{12}}{{C}_{5}}=\dfrac{12!}{(7)!5!} \\
 & {{\Rightarrow }^{12}}{{C}_{5}}=\dfrac{12\times 11\times 10\times 9\times 8}{5\times 4\times 3\times 2\times 1} \\
 & {{\Rightarrow }^{12}}{{C}_{5}}=792 \\
\end{align}\]

Therefore, the binomial coefficient of \[^{12}{{C}_{5}}\] is 792.

Note: While we are finding the factorial, it is not mandatory to multiply with zero as 0! =1 therefore it will not affect the final value. Keep in mind the formula used to find the factorial so that in future you can easily solve questions of this type. Perform the calculations carefully.