Increasing and Decreasing Numbers

[eddybob123]

Intended for high school students:
Let a positive integer be called "increasing" if every one of its digits is strictly greater than the one directly to the left of it. For example, 289 is increasing but 6788 is not.
How much 3 digit increasing numbers are there?
How much 4 digit increasing numbers are there?
How much n digit increasing numbers are there?
(Note: a number can't start with 0.)
Likewise, call a number "decreasing" if every digit is less than the digit directly to the left of it.
How much n digit decreasing numbers are there?

 

[Aaron Bex]

3 Digit Increasing Numbers:
There are 8 increasing 3-digit numbers, which are 123, 124, 125, 134, 135, 145, 234, and 235.

4 Digit Increasing Numbers:
There are 36 increasing 4-digit numbers, which are 1234, 1235, 1245, 1345, 2345, 1236, 1246, 1256, 1356, 1456, 2356, 2456, 1237, 1247, 1257, 1357, 1457, 2357, 2457, 3457, 1238, 1248, 1258, 1358, 1458, 2358, 2458, 3458, 1239, 1249, 1259, 1359, 1459, 2359, 2459, 3459, and3459.

n Digit Increasing Numbers:
For any n-digit increasing number, there will be (n-1) x 9 such numbers. For example, for a 5-digit increasing number, there will be 4 x 9 = 36 such numbers.

n Digit Decreasing Numbers:
For any n-digit decreasing number, there will be (n-1) x 9 such numbers. For example, for a 5-digit decreasing number, there will be 4 x 9 = 36 such numbers.