Puzzle #86: Add The Digits And Count

I found this beautiful puzzle on Brilliant.org very recently. A very different kind of puzzle, requires basic knowledge of combinatorics. I have also created a couple of simpler versions so that children can also attempt it.

Original Problem: How many integers between 1 and 1,000,000 have the sum of their digits equal to 18?

Variation 1: How many integers between 1 and 1,000 have the sum of their digits equal to 9?

Variation 2: How many integers between 1 and 1,000 have the sum of their digits equal to 18?

As always, please send your answers directly to me at alokgoyal_2001@yahoo.com. If you like the puzzle, please share it with others. If you have interesting puzzles to share, please send them to me at my e-mail given above.

Happy counting!

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