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juantheron

Member

Registered: 2011-10-19

Posts: 312

multinomial theorem

no of solns of x+y+z=15.My main is concern is how to solve it through multinomial theorem(Why and how it is used like that) please explain in detail

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: multinomial theorem

First you need with a diophantine equation to always have bounds on the variables. For your problem a reasonable one is:

0 <= x,y,z and x,y,z are integers.

Now to use the multinomial start by drawing 15 x's

x x x x x x x x x x x x x x x

Place two underscores anywhere between or around them like this:

x x x x x x _ x x x x x _ x x x x

Notice that the x's have been split into 3 groups. one has 6 x's, another has 5 x's and the last has 4 x's. Now 6 + 5 + 4 = 15. What you have done just by placing 2 underscores has been to come up with a solution to x+y+z=15 with x = 6 and y = 5 and z = 4.

Here is another

x_ x x _x x x x x x x x x x x x

you have another solution that says x = 1, y = 2 and z = 12.

Yet another

xxxx _ _ x x x x x x x x x x x

you have another solution that says x = 4, y = 0 and z = 11.

So solving that equation is equivalent to finding the number of arrangements of 15 x's and 2 underscores.

That is where the multinomial theorem comes in:

So there are 136 solutions to that diophantine equation with the above conditions.

You can do the calculation using generating functions like this:

See the coefficient of x^15 , it is 136.

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