Question 3491: A number is a decreasing number if it has two or more digits and each digit is less than the digit to its left. For example 754 and 620 are decreasing numbers, but 756 and 6220 are not decreasing numbers. How many decreasing numbers are there?
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! You did not mention that how many total digits in the decreasing number.
In fact,the longest possible decreasing integer is 9876543210. iT
contains 10 digits.
Now, I consider all possible cases:
(i)Ten digits: C(10,10) = 1. [There is only one number,namely 9876543210]
(ii)Nine digits: there are C(10,9) = 10 dec. numbers.
...
r digits, r>=2,: there are C(10,r) dec. numbers.
Totally, there are C(10,10)+C(10,9)+..+C(10,2) = 2^10 -C(10,1) -C(10,0)
= 1024 - 11 = 1013 such dec. numbers. (two or more digits)
Kenny
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