How many 4-digit numbers in the form $\overline { abcd }$ with $a\le b\le c\le d$ .

The answer is 495.

**
This section requires Javascript.
**

You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.

The digits we are going to use are $1, 2, 3, 4, 5, 6, 7, 8, 9$ .

For $a < b < c < d$ , we have

= ${ 9 \choose 4} = 126$

For $a < b < (c = d)$ , we have

= ${ 9 \choose 3} = 84$

For $a < (b = c) < d$ , we have

= ${ 9 \choose 3} = 84$

For $(a = b) < c < d$ , we have

= ${ 9 \choose 3} = 84$

For $a < (b = c = d)$ , we have

= ${ 9 \choose 2} = 36$

For $(a = b) < (c = d)$ , we have

= ${ 9 \choose 2} = 36$

For $(a = b = c) < d$ , we have

= ${ 9 \choose 2} = 36$

For $a = b = c = d$ , we have

= ${ 9 \choose 1} = 9$

The number of 4-digit numbers with $a \leq b \leq c \leq d$ are:

$= 126 + 84 + 84 + 84 + 36 + 36 + 36 + 9$

$= \boxed{495}$