50!

Question

How many digit number is 50! ?

1. 40
2. 65
3. 68
4. 72

kirankp · Answer

The logarithm of the factorial can be used to calculate the number of digits that the factorial number is :
 https://files.dropbox.com/u/35146/stackoverflow/log-factorial.png

or a better way to find out is using Stirling's approximation/Srinivasa Ramanujan approximation.

https://en.wikipedia.org/wiki/Factorial

but i guess there is no easier method than the above said to find the number of digits in a factorial and for GMAT it may be above 800 question

BTW number is 65.(source wiki) OA B

klusener · Answer

guys , is this is a gmat quant question.

the first thing that came to my mind was n! = n ( n - 1 ) ! .

But this algorithm doesn't give me the digits !!

Hussain15 · Answer

I dont think such a question can be asked on GMAT. Need expert opinion on this!! :?:

How to solve this question??

shatrughna · Answer

kirankp

please tell me how you reached to this answer? even after going thru wiki I am unable to understand

PussInBoots · Answer

This question will not be asked on GMAT. This question is impossible to solve w/o use of logarithms.

kirankp · Answer

to arrive at the answer u need a scientific calculator

u can't solve this by just manual calculation. here is the expanded form log 1 + log 2 + log 3+ ... +log 50; use log i the scientific cal to solve each log value and add it. PS: this is not a gmat question but if u want to know the value here is a much easier way to find : //Pseudo-code function factorialDigits (n) var result = 0; for(i = 1; i<=n; i++) result += log10(n); return result;

klusener · Answer

I saw the code for this on several programming sites and i can myself write a small program to do this .

But there is no manual way to solve this.

thanks,

shatrughna · Answer

I am in 12th and preparing myself for competitive exam. Found this sum in one book. Please tell me how do I prepare for these types of exam?

klusener · Answer

Mate if you have access to the log tables then you can sum the natural logs .

thanks,

jeckll · Answer

Please label this appropriately as 900-level question :lol:

shatrughna · Answer

Found Stirling's approximation

50! = sqrt(2* pi * 50) * ( 50 raise to 50) * ( e raise to -50) = S
log S = (1/2) * log 100 + (1/2) log pi + 50 log 50 - 50 log e
= (1/2) * 2 + (1/2) * log 3.14 + 50*log 50 - 50 * log 2.718
= 1 + (1/2) * ( 0.4972) + 50 * ( 1.6990) -50 * ( 0.4343 )
= 64.4846

S= 3.05 * (10 raise to 64)
hence the number has 64 digits

Thanks a lot for your help ..........

50!

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