<s>[INST] Number of solution set such that $a \geq b \geq c \geq d \geq 0$; $a, b, c, d \in \mathbb{Z}$ and $a + b + c + d = 10$ Given $(a, b, c, d)$ is a set of integers and $a \geq b \geq c \geq d \geq 0$. Find the number of solution sets for $a + b + c + d = 10$.
(This is problem from a competition, the answer key says its $23$, wherein my answer is $455$)
Solution
$x_1 + x_2 +1 + x_3 + 2 + x_4 + 3 = 10$
$x_1 + x_2 + x_3 + x_4 = 16$
$_{15}C_3$ = 455
 [/INST] The solutions are:
$$\begin{align}(d,c,b,a)=
&(0,0,0,10) \cdots (0,0,5,5) \Rightarrow 6 \\ 
&(0,1,1,8) \cdots (0,1,4,5) \Rightarrow 4 \\
&(0,2,2,6), (0,2,4,4) \Rightarrow 3 \\
&(0,3,3,4) \Rightarrow 1  \\
&(1,1,1,7) \cdots (1,1,4,4) \Rightarrow 4 \\
&(1,2,2,5), (1,2,3,4) \Rightarrow 2 \\
&(1,3,3,3) \Rightarrow 1 \\
&(2,2,2,4),(2,2,3,3) \Rightarrow 2 \end{align}$$
Hence, there are $23$ solutions.
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