Question

Compute:

1. $\dfrac{{10!}}{{3!7!}}$
2. $\dfrac{{8!}}{{\left( {6 - 4} \right)!}}$

Answer 1

In the given questions first we will multiply all factorial numbers with our given number down to$1$, write all given factorial numbers in this format. After that, we will solve the expression. In part $2$ first we will do the subtraction of the given numbers then we will write all factorial numbers with our chosen number after that solve it. Thus we will get the answer.

Complete step by step Answer:

1. Given that:
   $\dfrac{{10!}}{{3!7!}}$
   $\Rightarrow \dfrac{{10 \times 9 \times 8 \times 7!}}{{3 \times 2 \times 1 \times 7!}} \\\ \Rightarrow \dfrac{{10 \times 9 \times 8}}{{3 \times 2 \times 1}} \\\ \Rightarrow 120 \\\$

The answer to the first part is 120.

2. Given that:
   $\dfrac{{8!}}{{\left( {6 - 4} \right)!}}$
   In the given question first, we will subtract in the denominator.
   $\Rightarrow \dfrac{{8!}}{{2!}} \\\ \Rightarrow \dfrac{{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}} \\\ \Rightarrow 8 \times 7 \times 6 \times 5 \times 4 \times 3 \\\ \Rightarrow 20160 \\\$
   The answer for part $2$ is 20160

Note: In the given question we have to multiply all factorial numbers with our given number down to $1$, without this we cannot solve the given expression. In the second part of the given question remember that first, we have to do subtraction after that to solve the factorial. Thus we get the correct answer.