Counting number of integer solutions How many integer solutions are there to the equation $a + b + c = 21$ if $a \geq3$,
$b \geq 1$, and $c \geq 2b$?
If $a = 3+A$, $b =1+B$, $c = 2b + d=2B+d+2$, the equation is $A + 3 B + d = 15$ with $A, B, d \ge 0$.  Now for each $B$ from $0$ to $5$, 
the number of $(A,d)$ pairs is $16 - 3 B$.  So...
$a$ can vary from $3$ to $18$ $b$ can vary from $1$ to $16$ $c$ can vary from $2$ to $17$ Hence according to me number of solutions should be $16*16*16=4096$