The number of integers x, y, z, w satisfying | vedclass

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(B) Given the equation $x+y+z+w=25$ with constraints $x, y, z \geq -1$ and $w \geq 1$. Let $a = x+1 \geq 0$,$b = y+1 \geq 0$,$c = z+1 \geq 0$,and $d =

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${}^{30}C_3$

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(B) Given the equation $x+y+z+w=25$ with constraints $x, y, z \geq -1$ and $w \geq 1$. Let $a = x+1 \geq 0$,$b = y+1 \geq 0$,$c = z+1 \geq 0$,and $d = w-1 \geq 0$. Substituting these into the equation: $(a-1) + (b-1) + (c-1) + (d+1) = 25$. This simplifies to $a+b+c+d-2 = 25$,or $a+b+c+d = 27$. The number of non-negative integral solutions for $a+b+c+d = n$ is given by the formula ${}^{n+k-1}C_{k-1}$,where $k$ is the number of variables. Here $n=27$ and $k=4$,so the number of solutions is ${}^{27+4-1}C_{4-1} = {}^{30}C_3$.

The number of integers $x, y, z, w$ satisfying $x+y+z+w=25$ and $x, y, z \geq -1, w \geq 1$ is

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