Easy Combinatorics Problem

Level 2

Find the number of ordered quadruples of positive odd integers ( x 1 , x 2 , x 3 , x 4 ) (x_1, x_2, x_3, x_4) that satisfies the following equation

x 1 + x 2 + x 3 + x 4 = 98 x_1+x_2+x_3+x_4 = 98

18600 17600 19600 20600

Alan Yan by Alan Yan , 20, USA

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1 solution

Alan Yan
Sep 9, 2015

Let x i = 2 a i + 1 x_i = 2a_i + 1 where a i a_i is nonnegative.

The equation simplifies to a 1 + a 2 + a 3 + a 4 = 47 a_1+a_2+a_3+a_4 = 47 for nonnegative integers. This is easy, because it is just stars and bars, sticks and stones, chopsticks and bowls, river stones and branches, 1 and 0 , etc.

Thus, the answer is ( 50 3 ) = 19600 {50 \choose 3} = 19600

4 Helpful 0 Interesting 0 Brilliant 0 Confused

stars and bars, sticks and stones, chopsticks and bowls, river stones and branches, 1 and 0 , etc.

Best description ever.

Pi Han Goh - 5 years, 9 months ago

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Don't know whether that is sarcastic, but here you go if you want to learn more: http://artofproblemsolving.com/wiki/index.php/Ball-and-urn

Alan Yan - 5 years, 9 months ago

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