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Question: How do you evaluate \({}^{8}{{C}_{3}}\) ?...

How do you evaluate 8C3{}^{8}{{C}_{3}} ?

Explanation

Solution

In the given question we need to find the combination value for total 8 terms considering 3 of them. That means we need to find the number of combinations of group 3 from total 8 terms and need to simplify the last answer.

Complete step by step answer:
According to the given question we need to find 8C3{}^{8}{{C}_{3}}.
Now, we know that combination is basically a technique in which we determine the number of possible arrangements in a collection of items in which we don’t consider the pattern of arrangements. In the combination we are only interested in the arrangement not the order of the arrangement.
Now, we are given 8 different things which we need to arrange in all possible ways in a group of 3.
Now, we know that combination of n different elements in group of r we have the formula as nCr=n!(n−r)!r!{}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}.
And we also know that n!=n(n−1)....3.2.1n!=n\left( n-1 \right)....3.2.1
Now, in the given question we are given n=8n=8 and r=3r=3 .
Further, substituting these values in the given formula we get,
8C3=8!(8−3)!3! ⇒8×7×6×5!5!3×2 ⇒8×71! ⇒56 \begin{aligned} & {}^{8}{{C}_{3}}=\dfrac{8!}{\left( 8-3 \right)!3!} \\\ & \Rightarrow \dfrac{8\times 7\times 6\times 5!}{5!3\times 2} \\\ & \Rightarrow \dfrac{8\times 7}{1!} \\\ & \Rightarrow 56 \\\ \end{aligned}

Therefore, the value of 8C3{}^{8}{{C}_{3}} is equal to 56.

Note: In the given question we need to remember that we are not including the pattern of arrangement so we don’t need to include that which is very common mistake and also if we want to find the pattern of arrangement, we have the following formula as nPr=n!r!{}^{n}{{P}_{r}}=\dfrac{n!}{r!}