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# Distributions

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Find the number of solutions to

x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_10 <= 2

in nonnegagive integers.

Dec 16, 2022

#1
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If they sum to 2, there are 2 stars and 9 bars which makes for \({11 \choose 9} = 55\) solutions.

If they sum to 1, there is 1 star and 9 bars which makes for \({10 \choose 9} = 10\) solutions.

If they sum to 0, there are 0 stars and 9 bars which makes for \({9 \choose 9} = 1\) solution.

So, there are \(55 + 10 + 1 = \color{brown}\boxed{66} \) solutions.

Dec 16, 2022