Factorial First Or Second?

Number Theory Level 1

Which is greater:

$\large (100!)^2 \text{ or } (100^2)!$

Clarification : $!$ denotes the factorial notation.

(100!)^2

(100^2)!

They are the same

3 solutions

Aareyan Manzoor
Feb 12, 2016

$100!^2=1*2*3*4*....*100*1*2*3*....*100$ $100^2!=1*2*3*.....*100*101*102*....*200*x, x>1$ $100^2!>1*2*3*.....*100*101*102*....*200>1*2*3*4*....*100*1*2*3*....*100=100!^2$

RHS of the first line needs to be edited.

Devendra Kumar Singh - 5 years, 4 months ago

Why did you use asterisks instead of \times or \cdot? :D

A Former Brilliant Member - 5 years, 3 months ago

laziness :p

Aareyan Manzoor - 5 years, 3 months ago

Same method.

Damn it we'll have to call Dibs on solutions from now onwards :3

I was about to post the same solution.

Mehul Arora - 5 years, 4 months ago

You can always post an alternate solution with stirlings approx if you want :p

Aareyan Manzoor - 5 years, 4 months ago

Try This

Mehul Arora - 5 years, 4 months ago

Good Idea :P

Mehul Arora - 5 years, 4 months ago

Imran Ansari
Feb 18, 2016

$(100!)^2$ = $(100^2) !$

(100!) * (100!) = (10000)!

(100 * 99....1) * (100 * 99....1) = 10000 * 9999 * ..........1

10000 * 9801 *..........1 = 10000 * 9999 *.........1

When we simplify as shown above, the second term onward in the LHS the digits are coming lower than the RHS, you can see 9801 and 9999. So LHS will be lower

Use inequality sign instead of =

Arulx Z - 5 years, 2 months ago

Aastha Shah
Feb 13, 2016

try using induction

Factorial First Or Second?

3 solutions

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