Ways to get a sum

If the conditions are $a,b,c,d\geq0$ then by distributing into bins ,

https://brilliant.org/wiki/distinct-objects-into-distinct-bins/

The number of $4$ -tuple $(a,b,c,d)$ is just $\binom{14}{3}$ .

But we can make $a+b+c+d=15$ into $(a+2)+(b+3)+(c+4)+(d+5)=29$ which is also

$w+x+y+z=29$ where $w,x,y,z\geq0$ and $w=a+2, x=b+3, y=c+4, z=d+5$

Thus, by distributing into bins we have $\binom{28}{3}$ ways of choosing $(w,x,y,z)$ which is the same number of ways of choosing $(a,b,c,d)$

as $(w,x,y,z)=(a+2,b+3,c+4,d+5)$ which we can then find $(a,b,c,d)$ .

The question is equivalent to finding non-negative integers $(A,B,C,D)$ such that, $A + B + C + D = 25$ .

Separating $25$ similar objects into $4$ groups is equivalent to ordering $25$ balls and $3$ sticks, where the gaps between the sticks are the groups.

( eg O| | O O | O O represents $(1,0,2,2)$ )

This can be done $\binom{28}{3}$ ways.