Question

Find the number of ordered pairs $(x,y)$ of positive integer such that $x+y=1000$ and integer $x$ and $y$ have zero digit in it.

Answer 1

Given, $x+y=1000$.

Then $x=10,20,30,........,990$ and $y=990,980,970,.........,10$.

$\therefore$ no. of ordered pair $(x,y)$ will be, no. of terms in the series $10+20+30+.......+990$.

Here, this series is in A.P. with first terms $\left(a\right)=10$ and common difference $\left(d\right)=10$ and $n^{th}$ term $\left(a_n\right)=990$

Since, $a_n=a+(n-1)d$ [Where, $n$ is the number of terms in the given series.]

or, $990=10+(n-1)10$

or, $980=10(n-1)$

or, $n=99$

$\therefore$ total no. of order pairs $(x,y)=99$.