Tasty Factorials!

Number Theory Level 2

Find $x$

$\frac { { 6 }^{ 2 }!9!x! }{ 35!6!7!0! }=18144$

The answer is 7.

2 solutions

Ananya Srishti
Aug 12, 2015

$\quad \quad \frac { { 6 }^{ 2 }!9!x! }{ 35!6!7!0! } =18144\\ \Rightarrow \frac { { 36 }!9!x! }{ 35!6!7!0! } =18144\\ \Rightarrow \frac { { 36 }.9!x! }{ 10! } =18144\\ \Rightarrow \frac { { 36 }.x! }{ 10 } =18144\\ \Rightarrow \quad x!=5040\\ \therefore \quad x=7$

I did it til x! = 5040, and after that I calculated the factorial manually and found that 7! = 5040, is there any way to find the value of x given the value of x factorial?

Razif FA - 5 years, 9 months ago

even i did the same thing . i will check another method and tell u

Ananya Srishti - 5 years, 9 months ago

how did you know 7!6!=10!?

Yasir Soltani - 5 years, 9 months ago

because /( \frac { 7!6! }{ 10! } \quad =\quad 1\ herefore\quad 7!6!=10 /)

Ananya Srishti - 5 years, 9 months ago

Ikkyu San
Aug 12, 2015

$\begin{aligned}\dfrac{\color{#624F41}{(6^2)!}\ 9!\ x!}{35!\ 6!\ 7!\ 0!}=&\ 18144\\\dfrac{\color{#624F41}{36!}\ \color{#D61F06}{9!}\ x!}{35!\ 6!\ 7!}=&\ 18144\\\dfrac{\color{#624F41}{36\cdot\not{35!}}\cdot\color{#D61F06}{9\cdot8\cdot\not{7!}}\cdot x!}{\not{35!}\cdot\color{#302B94}{6!}\cdot\not{7!}}=&\ 18144\\\dfrac{36\cdot9\cdot8\cdot x!}{\color{#302B94}{720}}=&\ 18144\\\dfrac{18}5\cdot x!=&\ 18144\\x!=&\ 5040\\x=&\ \boxed7\end{aligned}$

i did it in a shorter way

Ananya Srishti - 5 years, 9 months ago

Tasty Factorials!

The answer is 7.

2 solutions

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