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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Permutation and Combination

Olympiad workbook · IOQM · Permutation and Combination

Consider the set S=\ (a, b, c, d, e): 0 < a < b < c < d < e < 100\...

Consider the set S={(a,b,c,d,e):0<a<b<c<d<e<100}S=\{(a, b, c, d, e): 0 < a < b < c < d < e < 100\} where a,b,c,d,ea, b, c, d, e are integers. If DD is the average value of the fourth element of such a tuple in the set, taken over all the elements of SS, find the largest integer less than or equal to DD.

Correct answer

66

Step-by-step solution

d=4 sum =4×95C1×3C3d=5 sum =5×94C1×4C3d=6 sum =6×93C1×5C3⋮d=98 sum =98×nC1×97C3\begin{array}{ll}d=4 & \text { sum }=4 \times{ }^{95} C_1 \times{ }^3 C_3 \\ d=5 & \text { sum }=5 \times{ }^{94} C_1 \times{ }^4 C_3 \\ d=6 & \text { sum }=6 \times{ }^{93} C_1 \times{ }^5 C_3 \\ \vdots & \\ d=98 & \text { sum }=98 \times{ }^n C_1 \times{ }^{97} C_3\end{array} Total =4⋅95C1⋅3C3+5⋅94C1⋅4C3…×981C1⋅97C3=4 \cdot{ }^{95} C_1 \cdot{ }^3 C_3+5 \cdot{ }^{94} C_1 \cdot{ }^4 C_3 \ldots \times 98{ }^1 C_1 \cdot{ }^{97} C_3 =4∑r=195(99−r)4⋅r98−rC3=4∑r=19599−rC4×r=4×100C6∴ A.M =4×100C699C5=2003∴[2003]=66\begin{aligned} & =4 \sum_{r=1}^{95} \frac{(99-r)}{4} \cdot r^{98-r} C_3 \\ & =4 \sum_{r=1}^{95}{ }^{99-r} C_4 \times r \\ & =4 \times{ }^{100} C_6 \\ \therefore \quad & \text { A.M }=\frac{4 \times{ }^{100} C_6}{{ }^{99} C_5} \\ & =\frac{200}{3} \\ \therefore \quad & {\left[\frac{200}{3}\right]=66 }\end{aligned}
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