Number of ways to enter 1,2,3,4,5,6 on the spots marked on three intersecting circles and have the sums of 14

Question

How to solve it , i applied so many method but i could not find right answer, suggest me some method combination question gre 44

Answer 1

HINT: You can easily form a matrix from your set of equations. You should eventually find that $a+c=e+f=b+d=7$. The rest should be straightforward.

Answer 2

Let $\;a+c=A$, $\;b+d=B$, $\;e+f=C$.

You have the equations

$$ A+B=B+C=C+A=14\\ A+B+C=21 $$

So $\;A=B=C=7$, and that means $\;a+c=e+f=b+d=7$.

That means you have only to decide which variables are $1,2,3$, and the others are set.