Stirling's Approximation for n!

titun

Which is greater among 100 ³⁰⁰ and 300 ! ??

Please give the solution & not just the answer.

I am waiting for a magnificient solution.

prathima

i think 300! is greater
in both the numbers it is the product of 300 numbers
in 300! 200 numbers r greater than the 200 no. in 100^300
only 100 no. r less
so i think 300! is greater than 100^300
do correct me if i m wrong

nano0101

100 ³⁰⁰
300 ! = 100! (300!/100!)
(300!/100!) has 200 terms which are greater than 100
=> (300!/100!) >100²⁰⁰ -------1
100! < 100¹⁰⁰ ----------2
300! > 100³⁰⁰ --------from 1 n 2

titun

Nano0101 u messed up. Observe the following steps:

3 > 2 ........... eqn. 1

1 < 5..............eqn. 2

Now u just cannot multiply eqns 1 & 2 to say that 3 > 10. It is not always true as u can see in the above e.g. The bottom line is that,

a > b

c < d

No one in the world can predict that ac > bd.

So ur step involving the following is not valid.

(300!/100!) >100²⁰⁰ -------1
100! < 100¹⁰⁰ ----------2
300! > 100³⁰⁰ --------from 1 n 2

But nice try yaar !!!

Now Prathima, I'm not convinced by ur method of approach but it was a good attempt altogether. Can u think of anything else ?

amaron

300! > 100³⁰⁰

This is a direct consequence of Stirling's approximation of factorial values.

Stirling's approximation for higher factorials states that

$n!\approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n.$

There is also another relation

(n/3)ⁿ < n! < (n/2)ⁿThat solves the Problem!

amar.gupta

good work amaron

edison

Use this approach

Stirling's Approximation for n!

When evaluating distribution functions for statistics, it is often necessary to evaluate the factorials of sizable numbers, as in the binomial distribution:

A helpful and commonly used approximate relationship for the evaluation of the factorials of large numbers is Stirling's approximation:

A slightly more accurate approximation is the following

but in most cases the difference is small. This additional term does give a way to assess whether the approximation has a large error.

Stirling's approximation is also useful for approximating the log of a factorial, which finds application in evaluation of entropy in terms of multiplicity, as in the Einstein solid. The log of n! is

but the last term may usually be neglected so that a working approximation is

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Stirling's Approximation for n!