A number is called a “decreasing number” if each digit in the number is less than the digit to its left. For example, 87420 is a decreasing number. How many five-digit decreasing numbers are there?

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24-03-2018
Mathematics

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A number is called a “decreasing number” if each digit in the number is less than the digit to its left. For example, 87420 is a decreasing number. How many five-digit decreasing numbers are there?

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sqdancefan sqdancefan · Answer 1 · 2018-03-24T23:04:07+01:00

There are 252 such numbers.

_____
You can count from the number of numbers with bottom digits as small as they can be. For example, there are ...
.. 6 numbers ending in 3210
.. 5 numbers ending in 4210
...
.. 1 number ending in 8210, for a total of 21 numbers ending in 210.

There are 6 fewer, so 15 numbers, ending in 310. Carrying this onward, we find 56 numbers ending in 10, and 126 numbers ending in 0.

Likewise, there are 70 numbers ending in 1, 35 numbers ending in 2, 15 numbers ending in 3, 5 numbers ending in 4, and 1 number ending in 5.

Thus the total of decreasing numbers is
.. 126 +70 +35 +15 +5 +1 = 252