Question

There are $4$ red, $5$ green, and $6$ blue balls inside a box. If $N$ number of balls are picked simultaneously, what is the smallest value of $N$ that guarantees there will be at least two balls of the same colour? One cannot see the colour of the balls until they are picked.

• A. 4
• B. 15
• C. 5
• D. 2

Hint:

When a problem asks for a guaranteed repetition among categories, apply the Pigeonhole Principle: pick one from each category first, then add one more to force a repeat.

Answer 1

The Correct Option is A

Step 1: Identify the categories (pigeonholes).
There are $3$ colours: Red, Green, Blue $\Rightarrow$ $3$ pigeonholes.

Step 2: Worst-case reasoning (Pigeonhole Principle).
To avoid getting two of the same colour as long as possible, pick one ball of each colour first.
After $3$ picks, it is still possible that all $3$ balls are of different colours.

Step 3: Force a repeat.
The next (4th) ball must match one of the already chosen colours, because only $3$ colours exist.
Therefore $N=4$ guarantees at least two balls of the same colour. \[ \boxed{N_{\min}=4} \]