math help please?

Question

A number is called a decreasing number if it has two or more digits and each digit is less than the digit to its left. For example, 7421, 964310, and 52 are decreasing numbers, but 3421, 6642, 8, and 963212 are not. how many decreasing numbers are there?

Answer 1

A heck of a lot! But as answerer #1 pointed out, there is a maximum decreasing number, therefore the number of decreasing numbers is not infinite.

I can't think of any slick way to easily calculate the number of decreasing numbers though, sorry! Though I think I'd try to tackle it by looking at 3-digit numbers, then 4-digit, etc., up to 10-digit numbers.

Good luck!

~ ♥ ~

Answer 2

The highest is 9876543210 so it is not infinite. Here's how to think of it.

Suppose there were only 1one digit. There would be no decreasing numbers since you need 2 digits.

If there were 2 digits 0 and 1, you'd have 1 number: 10

If there were 3 digits 0, 1, and 2, you'd have
one 3digit number and three two-digit numbers (21, 20, 10)

If there were four (0,1,2,3) you'd have:
1 four-digit
4 three digit (321,320,310,210)
6 two digit (32,31,30,21,20,10)

Finally if there were five digits (0,1,2,3,4) you'd have
1 five digit
5 four digit (4321,4320,4310,4210,3210)
10 three digit (432,431,430,421,420,410,321,320,310,210)
10 two digit (you find them)

You should see the rows Pascal's Triangle developing (minus the last 2 numbers in each row)

It is amazing how many things that triangle applies to!

So anyway, for 10 digits, continue the triangle until the 10 row.

Send an IM if you need more help with it.

Answer 3

As answerer 1 pointed out, there IS a highest number and the set is NOT infinite. But there are too many numbers and too little time for me to calculate here. Just give Answerer 1 10 points.

Answer 4

there are endless number
so there's an infinite set of decreasing numbers