I was reading Sheldon Ross for probability and found the below question and had a doubt in it.
Consider we have a set of $n$ items out of which m are (similar) and defective and the remaining $n-m$ are (also similar and) functional. Our objective is to determine the number of linear orderings in which each pair of defective items is separated by at least $2$ functional units.
They solved it in the below manner :
Let us denote $x_1$ as the number of functional items to the left of the first defective, $x_2$ as the number of functional items between the first two defectives, and so on.
Then we have the equation
$x_1 +x_2+....+x_{m+1}=n-m$ \, $x_1 \geq 0 ,x_{m+1} \geq 0 ,x_i\geq2 ,i=2,...,m$
Upon letting $y_1=x_1+1$, $y_i=x_i-1,i=2,...,m\,,y_{m+1}=x_{m+1}+1$
we see that this is same as the number of positive solutions of the equations
$y_1+....+y_{m+1}=n-2m+3$
I understood the equation $x_1 +x_2+....+x_{m+1}=n-m$ but after that, I am unable to get how the parameters from $x_1$ to $x_{m+1}$ have conditions and how using the change of variables they converted it into another form.
Please explain.
Here's the idea . . .
The equation $$x_1+\cdots +x_{m+1}=n-m$$ with integer variables, subject to the restrictions $$ \begin{cases} x_k\ge 0,\;\text{for}\;k\in\{1,m+1\}\\[4pt] x_k\ge 2,\;\text{for}\;2\le k \le m\\ \end{cases} $$ holds if and only if $$(x_1+1)+\Bigl((x_2-1)+\cdots+(x_m-1)\Bigr)+(x_{m+1}+1)=(n-m)-(m-1)+2$$ or equivalently, $$y_1+\cdots +y_{m+1}=n-2m+3$$ where $$ y_k= \begin{cases} x_k+1,\;\text{for}\;k\in\{1,m+1\}\\[4pt] x_k-1,\;\text{for}\;2\le k\le m\\ \end{cases} $$ and the only restrictions are that $y_1,...,y_{m+1}$ are positive integers.
The point is that the change of variables takes the nonnegative integer variables, and adds $1$ to each, forcing the new ones to be positive integers, and takes variables which are required to be at least $2$, and subtracts $1$ from each, so the new ones are only required to be positive integers.
Hence, after the change, the initial restrictions have been "leveled out", in the sense that the new variables are only required to be positive integers.