How many solutions of $ x+y+z+t = 7 $ I have an equation: $ x+y+z+t = 7$. I want to know how many solutions does this equation have? $x,y,z$, and $t$ are positive integers. I have no idea how to solve this. Can you please help me to solve this question?

This is just a hint to help you get started thinking about the problem. There might be a cleaver way to look at this, but you could also simply just count the number of solutions. So you start
$$
1 + 1 + 1 + 4 = 7 \\
1 + 1 + 2 + 3 = 7 \\
1 + 1 + 3 + 2 = 7 \\
1 + 1 + 4 + 1 = 7 \\
1 + 2 + 1 + 3 = 7 \\
1 + 2 + 2 + 2 = 7 \\
\vdots
$$
Maybe a slightly smarter way to do it would be first to notice that $x,y,z,t$ need to be from $\{1,2,3,4\}$. Then figure out how many ways you can add these up to get $7$. You get (Note for example that if one number is $3$, then the other numbers have to be $1$ or $2$):
$$
1, 1, 1, 4 \\
1, 1, 2, 3 \\
1, 2, 2, 2. \\
$$
If you pick $1,1,1,4$ then how many ways can you add these up?
