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Question 1150894: We call a number a descending number if each digit is strictly smaller than the digit that comes before it. For example, 863 is a descending number. How many 3-digit descending numbers are there?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
If 2 is in the hundreds place, then there is only 1 three-digit number that is a descending number and it is 210.
If 3 is in the hundreds place, then there are three such descending numbers
310
320
321
If 4 is in the hundreds place, then we have six descending numbers
410
420
421
430
431
432
If 5 is in the hundreds place, then we have ten descending numbers
510
520
521
530
531
532
540
541
542
543
The counts so far are: 1, 3, 6, 10
The jump from 1 to 3 is +2
The jump from 3 to 6 is +3
The jump from 6 to 10 is +4
This pattern continues because we have 15 descending numbers if 6 is the hundreds digit
610
620
621
630
631
632
640
641
642
643
650
651
652
653
654
And there are 15+6 = 21 descending numbers with 7 as the hundreds digit, 21+7 = 28 descending numbers with 8 as the hundreds digit, and finally 28+8 = 36 descending numbers with 9 as the hundreds digit.
The counts are
1,3,6,10,15,21,28,36
Add them up
1+3+6+10+15+21+28+36 = 120
Answer: 120
Below is a pastebin link of all 120 three-digit descending numbers
https://pastebin.com/WqFYtfc4
This is to avoid adding further clutter to this current page.
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