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From a class of 15 students, 10 are to chosen for a picnic. There are two students who decide that either both will join or none of them will join. In how many ways can the picnic be organized?

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Case 1: Both join 

In case both the students decide to join, 

13C8 × 2C2 ways = 13C8 = \(\frac{13!}{8!\space5!}=1287\) 

Case 2: In case none of them join, 

it will be 13C10 Ways 13C10 = \(\frac{13!}{3!\space10!}=286\)

Total number of cases are = 13C8 + 13C10 

1287 + 286 = 1573

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