Question

Number of natural numbers of not more than $1000$ which can be formed using the digits $5,6,7,8\&9$ if digits are not repeated are _______.

Answer 1

Given $x=1,y=2,z=3\dots 9\text{(7 ways)}y=3,z=4\dots 9\text{(6 ways)}⋮y=8,z=9\text{(1 ways)}$

$\therefore \text{Total no. of ways}=7+6+\cdots +1$

When $x=2,y=3,z=4\dots 9\text{(6 ways)}y=4,z=5\dots 9\text{(5 ways)}⋮y=8,z=9\text{(1 ways)}$

$\therefore \text{Total no. of ways}=6+5+\cdots +1$

When $x=6,y=7,z=8,9\text{(2 ways)}y=8,z=9\text{(1 ways)}$

$\therefore \text{Total no. of ways}=2+1$

When $x=7,y=8,z=9\text{(1 ways)}$

$\therefore \text{Total no. of ways}=1$

$\therefore \text{Required no. of ways}=(7+6+\cdots +1)+(6+5+\cdots +1)+\cdots +(2+1)+1=85$