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I run a book club with z people, not including myself. Every day, for 400 days, I invite 5 members in the club to review a book. What is the smallest positive integer z so that I can avoid ever having the exact same group of 5 members over all 400 days?

 Mar 30, 2024
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To avoid ever having the same group of 5 members review a book, we need to ensure there are enough total members (z) such that any group of 5 can be chosen without repeating a combination.

 

Here's the key idea:

 

We care about the number of distinct groups of 5 members we can form, not just the total number of possible selections.

 

Combinations vs. Permutations:

 

Combinations: Order doesn't matter (e.g., John, Mary, Sarah is the same group as Sarah, John, Mary).

 

Permutations: Order matters (e.g., John reviewing first is different from Mary reviewing first).

 

In this case, since the order the members review the book doesn't matter, we're interested in the number of combinations of 5 members we can choose from a group of z people.

 

Using Combinations Formula:

 

The number of ways to choose 5 members out of z for a book review can be calculated using the combinations formula:

 

nCr = n! / (r! * (n-r)!)

 

where:

 

n = Total number of members (z)

r = Number of members chosen for review (5 in this case)

 

Finding the Smallest z:

 

We want to find the smallest positive integer z such that the number of combinations of choosing 5 members (nCr) is greater than or equal to the total number of days (400). In other words:

 

nCr >= 400

 

Trial and Error with Combinations:

 

We can try different values of z and calculate the corresponding nCr using the formula. The smallest z that satisfies the condition will be our answer.

 

For z = 5 (5 choose 5): 1 (There's only one group possible - all 5 members) - not enough.

 

For z = 6 (6 choose 5): 6 (We can choose 5 members out of 6 in 6 ways) - still not enough.

 

For z = 7 (7 choose 5): 21 (We can choose 5 members out of 7 in 21 ways) - finally enough!

 

Therefore, the smallest positive integer z is 7.

 Mar 30, 2024

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