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How many solutions are there to the equation
u + v + w + x + y + z = 18
where u, v, w, x, y, and z are nonnegative integers, and x is at most 10?

 Dec 20, 2022
 #1
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Try some cases, x = 10.

U + v + w + y + z = 8. Stars and Bars (8 + 5 - 1) choose (5 - 1) = 12 choose 4 = 495

x = 9.

u + v + w + y + z = 9. Star and bars 9 + 5 - 1 choose 5 - 1 = 13 choose 4 = 715.

Etcetera until you reach case x = 0, u + v + w + y + z = 18, Stars and bars 18 + 4 choose 4 = 7315.

Our lowest case is 12 choose 4, and we add up all the way until 22 choose 4.

12 choose 4 + 13 choose 4 + 14 choose 4 ... + 21 choose 4 + 22 choose 4 = 32857 solutions.

 Dec 20, 2022

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