2022-01-11

) i) In a group of 100 people, each person picks a number from 97 to 115. What is the minimum number of people you need to be sure two of them have the same number?

2022-03-08

i) In a group of 100 people, each person picks a number from 97 to 115. What is the minimum number of people you need to be sure two of them have the same number?

Vasquez

Expert2023-04-22Added 669 answers

To solve this problem, we need to use the Pigeonhole Principle, which states that if we have n items and k containers, with $n>k$, then at least one container must have more than one item.

In this case, we have 100 people who each pick a number from 97 to 115, so we have 19 possible numbers $(115-97+1=19)$.

To be sure that at least two people have the same number, we need to have more people than the number of possible numbers. Specifically, we need to have enough people to fill up all 19 possible numbers, plus one additional person to ensure that there are at least two people with the same number.

Therefore, we need at least $19+1=20$ people to be sure that two of them have the same number.

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