Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Permutation and Combination

Olympiad workbook · IOQM · Permutation and Combination

A quadruple (a, b, c, d) of distinct integers is said to be...

A quadruple (a,b,c,d)(a, b, c, d) of distinct integers is said to be balanced if a+c=b+da+c=b+d. Let SS be any set of quadruples ( a,b,c,da, b, c, d ) where 1≤a<b<d<c≤201 \leq a < b < d < c \leq 20 and where the cardinality of SS is 4411. Find the least number of balanced quadruples in SS.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,

Correct answer

91

Step-by-step solution

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,
 Total balanced quadruple =4(2C2+3C2…9C2)+10C2=4(10C3)+10C2=4×10×9×86+45=52520C4−4411=434 For least balanced =525−434=91\begin{aligned} & \text { Total balanced quadruple }=4\left({ }^2 C_2+{ }^3 C_2 \ldots{ }^9 C_2\right)+{ }^{10} C_2 \\ & =4\left({ }^{10} C_3\right)+{ }^{10} C_2 \\ & =\frac{4 \times 10 \times 9 \times 8}{6}+45=525 \\ & { }^{20} C_4-4411=434 \\ & \text { For least balanced }=525-434=91 \end{aligned}
Practice Permutation and Combination on Quantrex Academy →

More from Permutation and Combination

All Permutation and Combination questions Full Permutation and Combination list All Olympiad workbook PYQs