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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computer exploration (spoiler)

| Comment 1 of 3

Start with

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

After cancellations and cross-multiplying you get

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

This is true whenever x-y = y-z = 1.

for z=0:1000

for y=z:1000

yz=factorial(y-z);

for x=y:1000

xy=factorial(x-y);

ratio=factorial(x-z)/(xy*yz);

if ratio>1.8 && ratio < 2.2

disp([x y z ratio])

end

tests for other solutions, but finds none.

Posted by Charlie on 2022-11-07 10:37:14

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