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# Counting

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Help I need to use combinations

I run a book club with n people, not including myself. Every day, for 100 days, I invite members in the club to review a book. What is the smallest positive integer n so that I can avoid ever having the exact same group of members over all 365 days?

Mar 5, 2023

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To avoid ever having the exact same group of members over all 365 days, we need to make sure that no two groups of members are the same.

On the first day, we can invite any combination of members from the book club, except for just one member (since that would be the same group of members every day). That means there are a total of 2^n - 1 possible groups of members on the first day.

On the second day, we need to make sure that we don't invite the same group of members as on the first day. That means we can choose from the remaining 2^n - 2 possible groups of members.

Continuing in this way, on the k-th day, we can choose from 2^n - k + 1 possible groups of members, since we have already used up k - 1 groups of members on the previous k - 1 days.

To avoid ever having the exact same group of members over all 365 days, we need to make sure that the product of these possible choices is greater than or equal to 365. That is,

(2^n - 1)(2^n - 2)...(2^n - 365 + 1) >= 365.

Since we want to find the smallest positive integer n that satisfies this inequality, we can start with n = 1 and check each value of n until we find the smallest one that works.

For n = 1, we have:

(2^1 - 1)(2^1 - 2)...(2^1 - 365 + 1) = 1 * 0 * ... * (-363) = 0 < 365.

Therefore, n must be greater than 1. For n = 2, we have:

(2^2 - 1)(2^2 - 2)...(2^2 - 365 + 1) = 3 * 2 * ... * (-363) = a negative number.

Therefore, n must be greater than 2. For n = 3, we have:

(2^3 - 1)(2^3 - 2)...(2^3 - 365 + 1) = 7 * 6 * 5 * ... * (-363) = a negative number.

Therefore, n must be greater than 3. For n = 4, we have:

(2^4 - 1)(2^4 - 2)...(2^4 - 365 + 1) = 15 * 14 * 13 * ... * (-363) = a negative number.

Therefore, n must be greater than 4. For n = 5, we have:

(2^5 - 1)(2^5 - 2)...(2^5 - 365 + 1) = 31 * 30 * 29 * ... * (-363) = a negative number.

Therefore, n must be greater than 5. For n = 6, we have:

(2^6 - 1)(2^6 - 2)...(2^6 - 365 + 1) = 63 * 62 * 61 * ... * (-363) = a negative number.

Therefore, n must be greater than 6. For n = 7, we have:

(2^7 - 1)(2^7 - 2)...(2^7 - 365 + 1) = 127 * 126 * 125 * ... * (-363) = a negative number.

Therefore, n must be greater than 7. For n = 8, we have:

(2^8 - 1)(2^8 - 2)...(2^8 - 365 + 1) = 255 * 254 * 253 * ... * (-363) = a negative number.

Therefore, n must be greater than 8. For n = 9, we have:

(2^9 - 1)(2^9 - 2)...(2^9 - 365 + 1) = 511 * 510 * 509 * ... * (-363) = a negative number.

Therefore, n must be greater than 9. For n = 10, we have:

(2^10 - 1)(2^10 - 2)...(2^10 - 365 + 1) = 1023 * 1022 * 1021 * ... * (-363) = a negative number.

Therefore, n must be greater than 10. For n = 11, we have:

(2^11 - 1)(2^11 - 2)...(2^11 - 365 + 1) = 2047 * 2046 * 2045 * ... * (-363) = a negative number.

Therefore, n must be greater than 11. For n = 12, we have:

(2^12 -

Mar 5, 2023