Pigeonhole Principle?

If you have more pigeons than pigeonholes (boxes), then at least one box must have more than one pigeon.

Example: 10 pigeons, 9 boxes. Even if you spread them out as much as you can, one box will have 2 pigeons. There's no way around it.

Now the formula in your image:

Minimum items $= k(r - 1) + 1$

Here:

• $k =$ number of boxes
• $r =$ how many items you want in at least one box, guaranteed

This tells you the smallest number of items you need, so that no matter how you spread them out, at least one box ends up with r items.

Quick example:

• $4$ boxes $(k = 4)$
• You want to guarantee some box has at least $3$ items $(r = 3)$

Plug in: $4(3 - 1) + 1 = 4(2) + 1 = 9$

So with 9 items, you are guaranteed at least one box has 3 or more. With only 8, it's possible to spread them so every box has just 2. That's why 9 is the magic number.