Given that:$X_{1}+X_{2}+X_{3}+X_{4}=20$ where $X_{1}\geq3,X_{2}\geq1,X_{3}\geq0,X_{4}\geq5 $
we can write like this :
$X_{1}+X_{2}+X_{3}+X_{4}=20$-------->$(1)$ where $X_{1}-3\geq0,X_{2}-1\geq0,X_{3}\geq0,X_{4}-5\geq0$
Let $X_{1}-3=Y_{1},X_{2}-1=Y_{2},X_{3}=Y_{3},X_{4}-5=Y_{4}$
$X_{1}=Y_{1}+3,X_{2}=Y_{2}+1,X_{3}=Y_{3},X_{4}=Y_{4}+5$
Put the value in the equation $(1),$
$Y_{1}+3+Y_{2}+1+Y_{3}+Y_{4}+5=20$ where $Y_{1}\geq0,Y_{2}\geq0,Y_{3}\geq0,Y_{4}\geq0$
$\Rightarrow Y_{1}+Y_{2}+Y_{3}+Y_{4}+9=20$ where $Y_{1}\geq0,Y_{2}\geq0,Y_{3}\geq0,Y_{4}\geq0$
$\Rightarrow Y_{1}+Y_{2}+Y_{3}+Y_{4}=11$------>$(2)$ where $Y_{1}\geq0,Y_{2}\geq0,Y_{3}\geq0,Y_{4}\geq0$
The Number of Integral Solution of Equation $=\binom{n+r-1}{r}$ where $n=$Number of Variables and $r=$Sum of the variables.
Here $n=4$ and $r=11$
$\Rightarrow\binom{4+11-1}{11}$
$\Rightarrow\binom{14}{11}$
$\Rightarrow\binom{14}{3}=364$