Question

How do you evaluate $^8{C_3}$ ?

Answer 1

In the given question, we have been given to calculate the value of a combination (combination as in permutations and combinations). We can easily do that if we know the formula of combination. We then simplify the factorials by cutting the common terms out. And that is going to give us the answer.

Formula Used:
We have to calculate the value of a combination using the formula,
$^n{C_m} = \dfrac{{n!}}{{\left( {n - m} \right)!m!}}$.

Complete step by step answer:
We have to calculate the value of $^8{C_3}$.
To do that, we put in the value in the formula,
$^n{C_m} = \dfrac{{n!}}{{\left( {n - m} \right)!m!}}$
We put $n = 8$ and $m = 3$
So, we have,
$^8{C_3} = \dfrac{{8!}}{{\left( {8 - 3} \right)!3!}} = \dfrac{{8!}}{{5! \times 3!}} = \dfrac{{8 \times 7 \times {6} \times {{5!}}}}{{{{5!}} \times {6}}} = 56$.

Hence, $^8{C_3} = 56$

Additional Information:
Here, we calculated the value of a combination. But, if we had a permutation, then we would have used the formula:
$^n{P_m} = \dfrac{{n!}}{{\left( {n - m} \right)!}}$.

Note: In the given question, we had to calculate the value of a combination. To do that, we just need to know the formula to calculate the value of a combination. Then we put in the values, evaluate by cutting the common factorials and we get our answer.