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Question

Question: How do you evaluate \(^{8}{{C}_{2}}\)?...

How do you evaluate 8C2^{8}{{C}_{2}}?

Explanation

Solution

In this question we have been given with the combination function which has the general formula as nCr^{n}{{C}_{r}}, where nn stands for the total number of objects and rr is the number of items which has to be selected from the total number of items nn. In combination the order does not matter. We will use the formula of combination which is nCr=n!r!(n−r)!^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}. We will substitute the values of nn and rr in the formula and simplify to get the required solution.

Complete step by step answer:
We have the expression given to us as:
⇒8C2{{\Rightarrow }^{8}}{{C}_{2}}
We can see that the above expression is in the form of combination formula which is nCr^{n}{{C}_{r}} where n=8n=8 and r=2r=2.
We know that nCr=n!r!(n−r)!^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!} therefore, on substituting the values of nn and rr in the formula, we get:
⇒8C2=8!2!(8−2)!{{\Rightarrow }^{8}}{{C}_{2}}=\dfrac{8!}{2!\left( 8-2 \right)!}
Now on simplifying the bracket, we get:
⇒8C2=8!2!(6)!{{\Rightarrow }^{8}}{{C}_{2}}=\dfrac{8!}{2!\left( 6 \right)!}
Now we know that 8!8! can be written as 8×7×6!8\times 7\times 6! therefore, on substituting it in the expression, we get:
⇒8C2=8×7×6!2!(6)!{{\Rightarrow }^{8}}{{C}_{2}}=\dfrac{8\times 7\times 6!}{2!\left( 6 \right)!}
On simplifying the expression, we get:
⇒8C2=8×72!{{\Rightarrow }^{8}}{{C}_{2}}=\dfrac{8\times 7}{2!}
We know that 2!2! can be written as 2×12\times 1 therefore, on substituting, we get:
⇒8C2=8×72×1{{\Rightarrow }^{8}}{{C}_{2}}=\dfrac{8\times 7}{2\times 1}
On simplifying the expression, we get:
⇒8C2=4×7{{\Rightarrow }^{8}}{{C}_{2}}=4\times 7
On multiplying the terms, we get:
⇒8C2=28{{\Rightarrow }^{8}}{{C}_{2}}=28, which is the required solution.

Note: In this question we have used the combination formula. There also exists a permutation formula which is represented as nPr^{n}{{P}_{r}}, where nn is the total number of items and rr is the number of items which are to be selected from the total items. The formula for permutation is nPr=n!(n−r)!^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}. The important disjunction between permutation and combination is that the order of the items selected matters in permutation and not in combination.