Solve the problem. An average library contains at least 50 and at most 250 books. How many library owners must be polled to be certain that at least two owners have the same number of books in their libraries?
there are 201 numbers from 50 to 250.
So, clearly, if 202 people are polled, at least two of them must have the same number.
Oh no, it seems the library owners are playing a game of "Book Bingo"! To find out how many library owners need to be polled to ensure that at least two owners have the same number of books, let's put on our problem-solving glasses.
Since the minimum number of books in a library is 50 and the maximum is 250, we have a range of 201 different possible numbers of books (250 - 50 + 1 = 201).
To guarantee that at least two owners have the same number of books, we need to consider the worst-case scenario. In this scenario, each owner has a different number of books. The first owner has 50 books, the second has 51, the third has 52, and so on.
So, to ensure that no two owners have the same number of books, we need to poll at least 201 + 1 owners. That way, everyone would have a different number of books.
But if we want to guarantee that at least two owners have the same number of books, we need to poll just one more owner. So, the final answer is 201 + 1 + 1 = 203.
Therefore, you need to poll at least 203 library owners to be certain that at least two owners have the same number of books in their libraries. Good luck with your polling adventure!
To determine the minimum number of library owners that need to be polled, we can use the pigeonhole principle.
Step 1: Calculate the number of possible different numbers of books in a library.
Since the library must have at least 50 books and at most 250 books, there are a total of 201 possible numbers of books (250 - 50 + 1).
Step 2: Determine the number of pigeonholes.
In this case, the pigeonholes represent the possible numbers of books in a library.
Step 3: Determine the number of pigeons.
The pigeons represent the number of library owners that need to be polled.
Step 4: Use the pigeonhole principle.
According to the pigeonhole principle, if the number of pigeons (library owners) is greater than the number of pigeonholes (possible numbers of books), then at least two library owners must have the same number of books.
Therefore, we need to find the smallest number of library owners (pigeons) that is greater than the number of possible numbers of books (pigeonholes).
From step 1, we know that there are 201 possible numbers of books. To guarantee that at least two owners have the same number of books, we need to find the smallest number of library owners that is greater than 201.
Therefore, the minimum number of library owners that need to be polled to be certain that at least two owners have the same number of books is 202.
To solve this problem, we need to determine the minimum number of library owners that need to be polled to be certain that at least two owners have the same number of books in their libraries.
To find the minimum number of owners required, we can consider the extreme scenarios where each owner has either the minimum or maximum number of books.
For the minimum number of books, we have the range of 50 books. So, there are 50 possibilities for the number of books in the first owner's library. As we poll more owners, we can find at most 250 different numbers of books.
Now, let's calculate the number of owners needed to have more possibilities (numbers of books) than the range of 50 to 250.
To find the minimum number of owners, we want to guarantee that at least two owners have the same number of books. This is equivalent to finding the smallest number of owners such that the number of possibilities is greater than the range of 50 to 250.
The largest number of owners we can have is 250, as each owner must have a unique number of books.
Let's calculate the number of possibilities for a given number of owners. If we have n owners, the number of possibilities will be given by the equation 250 - n + 1 (as the number of possibilities decreases by 1 for each owner).
We want the number of possibilities to be greater than the range 50 to 250, so we have the inequality 250 - n + 1 > 250 - 50.
Simplifying the inequality, we get:
250 - n + 1 > 200.
Removing the unnecessary terms, we have:
-n + 1 > -50.
Simplifying further, we get:
1 > -50 + n.
Now, isolate the variable 'n':
n > 1 + 50.
Performing the addition:
n > 51.
Since the number of owners must be a whole number, the smallest number of owners required to guarantee two owners with the same number of books is 52.