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Is there a formula for this?

 

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.

 Jan 19, 2023
 #1
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Yes there definitely is a good strategy. We have 10 different variables.

Case 1, the sum is = 0:

There is 1 solution, where all variables are equal to 0.

Case 2, the sum is = 1:

There are 10 choose 1 ways to arrange the value of 1 among the 10 variables, so there are 10 solutions.

Case 3, the sum is = 2.

We can have 10 ways to arrange the 2. But if we have two 1s, we have (10 choose 2) ways to do it = 45.

 

Total sum is 10 + 45 + 10 + 1 = 66 total cases.

 Jan 19, 2023

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