Question

The total number of six digit numbers $x_1{x}_2{x}_3{x}_4{x}_5{x}_6$ having the property $x_1<x_2\le x_3<x_4<x_5\le x_6,$ is equal to:

Answer 1

$1\le x_1<x_2\le x_3<x_4<x_5\le x_6\le 9$

From the given conditions, it is clear that the first digit must be 1 or higher & the last digit must be 9 or lower.

Now imagine we have 8 balls each labelled with $+1$ sign.

Finally, imagine each number to be a divider

$1\le x_1<x_2\le x_3<x_4<x_5\le x_6\le 9$ ...|....|...|...|...|.....|.......

We have to distribute those 8 balls into the spaces between the dividers. This will tell us how much to add to the prior digit to get the next digit..

But first we have to put a $+1$ ball under each of the $<$ signs, because the difference between those numbers has to be at least 1.

$1\le x_1<x_2\le x_3<x_4<x_5\le x_6\le 9$ ...|0...|...|0...|0..|.....|.......

That leaves 5 balls we need to place between the 6 dividers.