Question

A package of baseball cards contains \({\bf{20}}\) cards. How many packages must be purchased to ensure that two cards in these packages are identical if there are a total of \({\bf{550}}\) different cards?

Short answer

We'll need at least \(27.55\) or \(28\) packets.

Step-by-step solution

Definition of Principle of pigeonholing

Principle of pigeonholing There is at least one box containing two or more objects if\(k\)is a positive integer and\(k + 1\)or more objects are arranged into\(k\)boxes.

Step 2: Solution

There are\(550\)different cards to choose from. According to the pigeonhole principle, we'll need at least\(551\)different cards, with at least two of them being similar.

Each bundle contains\(20\)cards:

\(\frac{{551}}{{20}} = 27.55\)

As a result, we'll require at least \(27.55\) packages or \(28\) packages (as we can only have an integer amount of packages).