Question

How do you evaluate ${}^{6}{{C}_{4}}$?

Answer 1

From the given question we have to evaluate the ${}^{6}{{C}_{4}}$. As we know that the combination means that the number of combinations of n objects taken r at a time is determined by the formula ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$. By expanding these factorials, we will get the solution.

Complete step by step answer:
From the question given we have been asked to evaluate the
$\Rightarrow {}^{6}{{C}_{4}}$
As we know that the combination means that the number of combinations of n objects taken r at a time is determined by the formula
$\Rightarrow {}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$
By comparing here, we will get the
$\Rightarrow n=6$
$\Rightarrow r=4$
By substituting in the formula, we will get,
$\Rightarrow {}^{6}{{C}_{4}}=\dfrac{6!}{4!\left( 6-4 \right)!}$
By simplifying further, we will get
$\Rightarrow {}^{6}{{C}_{4}}=\dfrac{6!}{4!\left( 2 \right)!}$
As we know that expansion of factorial,
$\Rightarrow n!=n\times \left( n-1 \right)\ldots \times 2\times 1$
From this we will expand the above,
$\Rightarrow {}^{6}{{C}_{4}}=\dfrac{6!}{4!\left( 2 \right)!}$
$\Rightarrow {}^{6}{{C}_{4}}=\dfrac{6\times 5\times 4!}{4!2}$
By further simplification we will get,
$\Rightarrow {}^{6}{{C}_{4}}=15$

Therefore, the evaluation of ${}^{6}{{C}_{4}}$ is $15$.

Note: Students should know the formula of combination. Students should be aware of formulas of permutation and combination. the number of combinations of n objects taken r at a time is determined by the formula ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$. the number of permutations of n objects taken r at a time is determined by the formula ${}^{n}{{p}_{r}}=\dfrac{n!}{\left( n-r \right)!}$.