Math Is Fun Forum

yakuza52

Member

Registered: 2008-09-01

Posts: 2

Combinatorics - How to Find an Unknown

I've been doing some work on combinatorics recently, including combinations and permutations.

My question relates to a combination problem where the number of objects to be picked and the total number of possible combinations are known, but not the number of objects that are chosen from.

For example, if three objects are chosen from a total of n objects and there are 56 possible combinations, what is the value of n?

I know the answer but I was just wondering if anyone would be able to discuss the methodology used to figure that value out.

Thanks in advance,

Ben

gnitsuk

Member

Registered: 2006-02-09

Posts: 121

Re: Combinatorics - How to Find an Unknown

For low numbers trial and error would be fine, in the example you gave we know that we need to use the formula:

Specifically we need to solve:

for n.

We can simply try integral values of n {1,2,3.....} and we soon arrive at n=8 as our solution.

In general this function will not have an integral solution.

The generalised factorial function is the Gamma function which defines the factorial of real and complex values: http://en.citizendium.org/wiki/Gamma_function

It is the inverse of this function which one would need in general - not a trivial thing. Google "inverse Gamma function" to find out more!

Last edited by gnitsuk (2008-09-02 20:55:02)

yakuza52

Member

Registered: 2008-09-01

Posts: 2

Re: Combinatorics - How to Find an Unknown

So there's no way you know of to solve the question using theory or formulas involving combinatorics itself?

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Combinatorics - How to Find an Unknown

First, the binomial coefficients have their combinatorial significance, so actually, what you're doing may seem only algebraical, but it's combinatorial. For small numbers (like 56) it's really most prefferable to use trial and error, because it will lead to solution faster. I can think of some kind of optimisation, where you don't actually try all the numbers. It may be used to tell if some number is binomial. There's a formula for the order (the maximal degree of a prime number p) which divides n!, that is:

We may also take advantage of it by factoring our number into prime divisors. In this case,
56=7*8
From this and the formula, we may deduce

, which, accidently leads to solution instantly, else you may play arranging the divisors in some way.

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