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Home » Surface area \( S \) of a sphere is \( 4\pi r^2 \). - High Altitude Science

Surface area \( S \) of a sphere is \( 4\pi r^2 \). - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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Let $ y_i = x_i - 1 $, so $ 0 \leq y_i \leq 5 $, and the equation beco...
y_1 + y_2 + y_3 + y_4 = 10
We now count the number of non-negative integer solutions to this equa...
\binom{10 + 4 - 1}{4 - 1} = \binom{13}{3} = 286
Now subtract the number of solutions where at least one $ y_i > 5 $. L...
Let $ A_i $ be the set of solutions where $ y_i \geq 6 $. Set $ z_i =...
|A_i| = \binom{4 + 4 - 1}{4 - 1} = \binom{7}{3} = 35
There are 4 such sets, so total for single violations: $ 4 \cdot 35 =...
Now consider intersections $ A_i \cap A_j $, where two variables $ \ge...
So total valid solutions:

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