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There are $k$ people in a room, each person picks a day of the year to get a free dinner at a fancy restaurant. $k$ is such that there must be at least one group of six people who select the same day. What is the possible value of such $k$ if the year is a leap year $(366$ days)?

  1. $1465$
  2. $1831$
  3. $1830$
  4. $2197$
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By Generalized Pigeonhole Principle,

If $k$ is a positive integer and $\text{N}$ objects are placed into $k$ boxes, then at least one of the boxes will contain $\left \lceil \frac{\text{N}}{k} \right \rceil$ or more objects.

Here, we need to find $\text{N,}$ and $k=366,$ and $\left \lceil \frac{\text{N}}{k} \right \rceil = 6$

Least value of $\text{N}$ that will satisfy this is $1831.$
Answer:

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