1. ## factorials

Solve for N:

7!8!9!/7!+8!+9!=8!N!/9

2. ## Re: factorials

Originally Posted by rubytuesday
Solve for N:

7!8!9!/7!+8!+9!=8!N!/9
noting the lack of parentheses in your equation, do you really mean this ...

$\displaystyle \frac{7! \cdot 8! \cdot 9!}{7! + 8! + 9!} = \frac{8! \cdot n!}{9}$

???

3. ## Re: factorials

Hello, rubytuesday!

$\displaystyle \text{Solve for }N:\;\;\frac{7!\,8!\,9!}{7!+8!+9!} \:=\:\frac{8!\,N!}{9}$

We have: .$\displaystyle \frac{8!\cdot N!}{9} \;=\;\frac{7!\cdot8!\cdot9!}{7!+8!+9!}$

. . Note that: .$\displaystyle 7! + 8! + 9! \;=\;7!(1 + 8 + 8\!\cdot\!9) \;=\;7!(81)$

So we have: .$\displaystyle \frac{8!\cdot N!}{9} \;=\;\frac{\rlap{//}7!\cdot 8!\cdot 9!}{\rlap{//}7!\cdot 9^2} \;=\;\frac{8!\cdot9!}{9\cdot 9} \;=\; \frac{8!\cdot(8!\cdot\rlap{/}9)}{9\cdot \rlap{/}9}$

. . . . . . . . . . $\displaystyle \frac{8!\cdot N!}{9} \;=\;\frac{8!\cdot 8!}{9}$

Multiply by $\displaystyle \frac{9}{8!}\!:\;\;N! \;=\;8! \quad\Rightarrow\quad \boxed{N \:=\:8}$

Thank You!