<s>[INST] How many non-negative integer solutions are there for $a+b+c+d=25$ if $a\geq 1, b\geq 2,c\leq 6,d\leq 14$ 
How many non-negative integer solutions are there for $a+b+c+d=25$ if $a\geq 1, b\geq 2,c\leq 6,d\leq 14$

So first I let $x= a-1$, $y=b-2$ and get:
$x+y+c+d=22$
And if all are non-negative I get that there are ${n+k-1\choose k-1} ={25\choose 3}$ solutions
Then I need to subtract the solutions where $c\geq 7,d\geq 15$
Let $z= c-7$ then $x+y+z+d=15$ and there are ${18\choose 3}$ non negative solutions
Let $w= d-15$ then $x+y+c+w=7$ and there are ${10\choose 3}$ non negative solutions.
And combining $z= c-7, w=d-15$ gives $x+y+z+w=0$, which will have only $1$ solution
So there are ${25\choose 3}-{18\choose 3}-{10\choose 3}-1$ solutions.
Does this seem correct?
 [/INST] Your solution is almost correct.
You subtract the case $"c\geq 7\; and\;d\geq 15"$ once in the case $c\geq 7$ and once in the case $d\geq 15$. So, you need to add the $1$ at the end.
Here for reconfirmation of the result the same calculation using generating functions:
$$[x^{22}]\frac{1}{1-x}\cdot\frac{1}{1-x}\cdot\frac{1-x^7}{1-x}\cdot\frac{1-x^{15}}{1-x} = [x^{22}]\frac{1-x^7-x^{15}+x^{22}}{(1-x)^4}$$
$$=[x^{22}](1-x^7-x^{15}+x^{22})\sum_{n \geq 0}\binom{n+3}3x^n$$
$$=\binom{25}3 - \binom{18}3-\binom{10}3 + \binom{3}3$$
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