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Counting problem with a few restrictions (combinatorics)

Let's take Keno for example:

From number 1 to 80, there are \(\dbinom{80}{20} = 3,535,316,142,212,174,320\) ways the casino can draw 20 numbers out of 80.

Given that the sum of the 20 numbers drawn is \(n\), how many combinations satisfy the following conditions:

  • \(x_1+x_2\ldots+x_{20} = n\)
  • \(x_1,x_2,\ldots,x_{20}\) are distinct
  • \(1 \leq x_i \leq 80 \)

Eg. If \(n = 210\), which is the minimum sum, \(1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20 = 210\), therefore there is only one possible combination for the sum of 20 numbers to be 210.

What about, say, \(n = 896\)? I have tried stars and bars but not really sure how to apply the distinct restriction.

Note by Lim Sy
1 week, 1 day ago

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Comment deleted 3 days ago

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edited, sorry sometimes even i get confused by my own question

Lim Sy - 5 days ago

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