A quadruple (a, b, c, d) of distinct integers is said to be balanced if a+c=b+d . Let S be any set of quadruples ( a, b, c, d ) where 1 a

Question

A quadruple (a, b, c, d) of distinct integers is said to be balanced if a+c=b+d . Let S be any set of quadruples ( a, b, c, d ) where 1 a < b < d < c 20 and where the cardinality of S is 4411. Find the least number of balanced quadruples in S .

Quantrex Academy · Accepted Answer

Correct answer: 91

aligned & Total balanced quadruple =4 ( ^2 C₂+ ^3 C₂ ^9 C₂ )+ ¹⁰ C₂ & =4 ( ¹⁰ C₃ )+ ¹⁰ C₂ & = 4 10 9 8 6 +45=525 & ²⁰ C₄-4411=434 & For least balanced =525-434=91 aligned

A quadruple (a, b, c, d) of distinct integers is said to be balanced if a+c=b+d . Let S be any set of quadruples ( a, b, c, d ) where 1 a < b < d < c 20 and where the cardinality of S is 4411. Find the least number of balanced quadruples in S .

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