Number

An increasing number has its digits increase in size from left to right. For example, 24589 is an increasing number. How many four-digit increasing numbers are there? 

There are 10C4 ways to choose 4 digits that is 210 ways

Now for each of these combiations there is exactly 1 where the digits are in ascenting order.

So that is still 210 ways.

If you discount numbers beginning with 0 then that gets harder.

Lets see... if the first one is 0 how many ways can the others be chosen. 9C3=84 ways

So I think there are 210-84= 126 ways

So there are 126 numbers that meet this criterion.

Here's another approach    ......

1 in 1st position  and then  -  C(8,3)   -   56

2 in 1st position and then - C (7,3)  -      35

3 in 1st position and then - C (6,3)  -      20

4 in 1st position and then  -  C (5,3)  -    10

5 in 1st position and then - C ( 4,3)  -       4

6 in 1st position and then  - C (3,3)  -       1

                                                              _____

                                                               126

Just as Melody found !!!

Since zero is not allowed at the beginning of each set of 4 ascending numbers, then the total number of 4-digit sets would simply be: 9C4 =126.