I run a book club with $n$ people, not including myself. Every day, for $365$ days, I invite five 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 five members over all $365$ days?

11 members.

Step-by-step explanation:

When we have a number N of objects and we want to divide it into groups K objects, the number of possible combinations is:

Here we have that C must be at least, 365 and K = 5, we want to find the value of N.

Now, you can play with different numbers of N, for example, if we take N = 9 we have:

C = 9!/(4!*5!) = 9*8*7*6/(4*3*2) = 126

so 9 is not enough, let's see N = 10

C = 10!/(5!*5!) = 10*9*8*7*6/(5*4*3*2) = 252

10 is not enough, let's see with 11.

C = 11!/(6!*5!) = 11*10*9*8*7/(5*4*3*2) = 462

So you need at least 11 members in the club.

step-by-step explanation:

y = | -2(-2) |     →     y = | 4 |     →     y = 4       ⇒   (-2, 4)

y = | -2(-1) |     →     y = | 2 |     →       y =   2       ⇒   (-1, 2)    

y = | -2(0) |     →     y = | 0 |     →       y = 0         ⇒   (0, 0)

y = | -2(1) |   c →     y = | -2 |     →     y =   2         ⇒   (1, 2)    

y = | -2(2) |     →     y = | - 4 |     →     y =   4         ⇒   (2, 4)