simplify the following factorial expressions into a multiplication statement. example: $\frac{5!}{2!cdot3!}$…

simplify the following factorial expressions into a multiplication statement. example: $\frac{5!}{2!cdot3!}$ can be written as to $\frac{5cdot4cdot3cdot2cdot1}{(2cdot1)cdot(3cdot2cdot1)}$ and then simplified to the multiplication statement $5cdot2$. note: hit the * button on your keyboard for multiplication. $\frac{7!}{3!cdot2!}=$ $\frac{8!}{2!cdot2!cdot2!}=$ $\frac{10!}{4!cdot3!cdot2!}=$

simplify the following factorial expressions into a multiplication statement. example: $\frac{5!}{2!cdot3!}$ can be written as to $\frac{5cdot4cdot3cdot2cdot1}{(2cdot1)cdot(3cdot2cdot1)}$ and then simplified to the multiplication statement $5cdot2$. note: hit the * button on your keyboard for multiplication. $\frac{7!}{3!cdot2!}=$ $\frac{8!}{2!cdot2!cdot2!}=$ $\frac{10!}{4!cdot3!cdot2!}=$

Answer

Explanation:

Step1: Expand factorials

Recall (n!=n\times(n - 1)\times\cdots\times1). So, (\frac{7!}{3!\times2!}=\frac{7\times6\times5\times4\times3!}{3!\times2\times1}).

Step2: Cancel out common terms

Cancel out (3!) in the numerator and denominator: (\frac{7\times6\times5\times4}{2\times1}=7\times6\times5\times2).

Step3: Expand second - expression factorials

For (\frac{8!}{2!\times2!\times2!}=\frac{8\times7\times6\times5\times4\times3\times2!}{2!\times2\times1\times2\times1\times2\times1}).

Step4: Cancel out common terms

Cancel out (2!) and simplify: (\frac{8\times7\times6\times5\times4\times3}{2\times2\times2}=8\times7\times6\times5\times3).

Step5: Expand third - expression factorials

For (\frac{10!}{4!\times3!\times2!}=\frac{10\times9\times8\times7\times6\times5\times4!}{4!\times3\times2\times1\times2\times1}).

Step6: Cancel out common terms

Cancel out (4!) and simplify: (\frac{10\times9\times8\times7\times6\times5}{3\times2\times2}=10\times9\times8\times7\times5).

Answer:

(7\times6\times5\times2) (8\times7\times6\times5\times3) (10\times9\times8\times7\times5)