Understanding Ball Picking with Guarantees

The question asks for the smallest number of balls, denoted as \(N\), that must be picked simultaneously from a box containing balls of different colors. The goal is to guarantee that at least two balls of the same color are obtained. It is important to note that the color of the balls cannot be seen until they are picked.

Pigeonhole Principle Application for Ball Picking

This problem is a classic application of the Pigeonhole Principle. This principle states that if you have more items than categories (or "pigeonholes"), at least one category must contain more than one item.

• Colors Available (Pigeonholes): The different colors of balls act as our categories or "pigeonholes".
• Red balls: 4
• Green balls: 5
• Blue balls: 6
• There are 3 distinct colors (Red, Green, Blue) in the box.

We are looking for the smallest number of "items" (balls picked) that guarantees at least two "items" fall into the same "container" (i.e., at least two balls picked are of the same color).

Balls: Worst-Case Scenario for Guarantees

To guarantee that at least two balls are of the same color, we need to consider the worst-case scenario. The worst-case scenario involves picking as many balls as possible without yet achieving a pair of the same color. This means we would pick one ball of each available distinct color before any color repeats.

• First Ball Picked: We pick one ball. It could be Red. (At this point, we have 1 ball, no pair).
• Second Ball Picked: We pick a second ball. To avoid getting a pair, it must be a different color from the first, say Green. (Now we have 2 balls, 1 Red, 1 Green, still no pair).
• Third Ball Picked: We pick a third ball. To avoid getting a pair, it must be a different color from the first two, say Blue. (Now we have 3 balls, 1 Red, 1 Green, 1 Blue, still no pair).

After these three picks, we have successfully picked one ball of each of the 3 distinct colors present in the box. All 3 balls are of different colors.

Smallest Guaranteed Value of N for Same Color

Once we have picked one ball of each color (a total of 3 balls), any additional ball picked must result in a pair of the same color. This is because there are no new colors left to pick from which would allow us to continue picking distinct colors.

• If the 4th ball picked is Red, we will now have 2 Red balls.
• If the 4th ball picked is Green, we will now have 2 Green balls.
• If the 4th ball picked is Blue, we will now have 2 Blue balls.

Therefore, after picking 3 balls (one of each distinct color), the addition of just one more ball guarantees that we will have at least two balls of the same color. The specific counts of each color (4 red, 5 green, 6 blue) are irrelevant to this specific guarantee, as long as there is at least one ball of each color available.

The smallest value of \(N\) that guarantees at least two balls of the same color is calculated as:

\(N = (\text{Number of distinct colors}) + 1\)

\(N = 3 + 1\)

\(N = 4\)

Thus, picking 4 balls simultaneously from the box guarantees that you will have at least two balls of the same color.