Nice Numbers

[Steven G]

I am having a great deal of trouble getting a handle on this problem.

Let x be in N. We call x nice if it is bigger than 10 and its decimal representation x = (a_k, ... , a₁) has the property that
a_i+1 < a_i for all 1 < i < k. For example 34 = (3, 4), 126 = (1, 2, 6) are nice while 3 = (3), 76 = (7, 6) are not. How
many nice numbers bigger than 10^5 are there?

I realized the 6 digit numbers go from 123,456 to 456,789

The 7 digit numbers go from 1,234,567 to 3,456,789

8 digits from 12,345,678 to 23,456,789

9 digits from 123,456,789 to 123,456,789

I can count these but in the worst way possible. I was hoping for a cleaner way to do this problem.

Thanks!

 

[lev888]

I am having a great deal of trouble getting a handle on this problem.

Let x be in N. We call x nice if it is bigger than 10 and its decimal representation x = (a_k, ... , a₁) has the property that
a_i+1 < a_i for all 1 < i < k. For example 34 = (3, 4), 126 = (1, 2, 6) are nice while 3 = (3), 76 = (7, 6) are not. How
many nice numbers bigger than 10^5 are there?

I realized the 6 digit numbers go from 123,456 to 456,789

The 7 digit numbers go from 1,234,567 to 3,456,789

8 digits from 12,345,678 to 23,456,789

9 digits from 123,456,789 to 123,456,789

I can count these but in the worst way possible. I was hoping for a cleaner way to do this problem.

Thanks!

For 6 digit numbers: The number of combinations of 6 objects from a set with 9 objects?

 

[Steven G]

For 6 digit numbers: The number of combinations of 6 objects from a set with 9 objects?

I'm sorry but I do not see why. Can you please explain?

 

[lev888]

I'm sorry but I do not see why. Can you please explain?

From http://home.ubalt.edu/ntsbarsh/Business-stat/otherapplets/ComCount.htm:
The number of combinations of k objects from a set with n objects is _nC^k. For example, the combinations of {1,2,3,4} taken k=2 at a time are {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, for a total of 6 = 4! / [(2!)(4-2) !] subsets.

The pairs in their example look suspiciously similar to your nice numbers.

 

[Steven G]

No comment. You are right. I should go to the corner for 123,456,789 hours!