perplexus.info :: Numbers : Binomial Coefficient Product Puzzle

Binomial Coefficient Product Puzzle (Posted on 2022-11-07)

Consider three positive integers x, y, and z that satisfy this equation:

comb(x,y)*comb(y,z) = 2*comb(x,z)

Does the above equation possess a finite number of solutions?

If so, determine all possible solutions of the above mentioned equation.

If not - then derive, with proof, the possible relationship between x, y, and z.

Note: comb(m,n) is the number of ways of choosing n unordered outcomes from m possibilities and defined as:

                m!
comb(m,n) = ---------
             n!(m-n)!

It is also known as binomial coefficient and read as "m choose n".

No Solution Yet Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Solution

| Comment 2 of 3 |

We have C(x,y) * C(y,z) = 2 * C(x,z). so lets start by substituting the definition:

x!/(y!(x-y)!) * y!/(z!(y-z)!) = 2 * x!/(z!(x-z)!)

Now move everything except the constant 2 to the left side and simplify:

(x-z)!/((x-y)!(y-z)!) = 2

The left side is just C(x-z, y-z), or equivalently C(x-z, x-y).

The only way for C(a,b) to equal 2 is when a=2 and b=1. Then x-z=2 and y-z=1.

So then x, y, z are three consecutive integers in descending order.

Posted by Brian Smith on 2022-11-07 11:36:40

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