This question contains formulas from my previous question The Number Of Integer Solutions Of Equations.
I understand the following thus far:
The number of distinct positive integer-valued vectors $(x_1,x_2,...,x_r)$ such that $$x_1 + x_2 + ... + x_r = n$$ is equal to $n-1\choose r-1$.
The number of distinct non-negative integer-valued vectors $(x_1,x_2,...,x_r)$ such that $$x_1 + x_2 + ... + x_r = n$$ is equal to $n+r-1\choose r-1$.
Problem 1
Suppose I want to count the number of distinct integer-valued vectors of the form $(x_1, x_2,...,x_r)$ such that $$x_1 + x_2 + ... + x_r = n$$
But now there are constraints involved $$x_1 \geq 0$$ $$x_r \geq 0$$ $$x_i > 0, i = 2,...,r-1 $$
In words: the first and last integers are greater or equal to zero and all in-between must be greater than zero.
How many such distinct vectors are possible?
My Approach To Problem 1
Let $$y_1 = x_1 + 1$$ $$y_i = x_i, i = 2,...,r-1$$ $$y_r = x_r + 1$$
This gives the positive integer-valued vector of the form $(y_1, y_2, ... , y_r)$ such that $$y_1 + y_2 + ... + y_r = n + 2$$
By (1), the number of distinct integer-valued vectors is equal to $n+2-1\choose r-1$ $=$ $n+1\choose r-1 $
Problem 2
Suppose I want to count the number of distinct integer-valued vectors of the form $(x_1, x_2,...,x_r)$ such that $$x_1 + x_2 + ... + x_r = n$$
The constraints involved are $$x_1 \geq 0$$ $$x_i \geq 2, i = 2,...,r-1 $$ $$x_r \geq 0$$
In words: the first and last integers are greater or equal to zero and all in-between are greater or equal to two.
How many such distinct vectors are possible?
My Approach To Problem 2
Let $$y_1 = x_1 + 1$$ $$y_i = x_i + 2, i = 2,...,r-1$$ $$y_r = x_r + 1$$
This gives the positive integer-valued vector of the form $(y_1, y_2, ... , y_r)$ such that $$y_1 + y_2 + ... + y_r = n + 2 + 2(r-2) = n + 2r-2$$
By (1), the number of distinct integer-valued vectors is equal to $n+2r-2-1\choose r-1$ $=$ $n+2r-3\choose r-1 $
I would appreciate it if you could verify my solutions and provide hints/suggestions.
