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 classic stars and bars combinatorics question.
We need to find 4 numbers, i.e. 4-tuples of positive integers such their sum is 7. So the number of solution is:
$$\binom{k-1}{n-1}$$
Where $k$ is their sum and $n$ is the the number of terms in the summation. So for our case we have:
$$\binom{7-1}{4-1} = \binom{6}{3} = 20 \text{ solutions}$$
Note that some of this solution are permutations because $(x,y,z,y) = (1,2,2,2)$ and $(x,y,z,y) = (2,1,2,2)$ are calculated twice. If you want solutions like these to be included just once, I can't think of a better way than counting. 
