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GATE 2023 CE (CE2) – Question 4

General Aptitude · Analytical Aptitude: Logical Reasoning and Deduction · 1 mark · Multiple choice

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.

  1. 4
  2. 15
  3. 5
  4. 2

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Show answer and explanation

Correct answer: (A) 4

Explanation

This is a pigeonhole-principle question. There are 3 colours (red, green, blue), and we want the smallest number of balls that guarantees two of one colour.

**Worst case:** the first three balls picked are all of different colours (one red, one green, one blue). This is possible, so $N=3$ does **not** guarantee a pair.

**Fourth ball:** whatever colour it has, it matches one of the three already picked, so a pair of the same colour is certain.

$$N_{min}=3+1=\mathbf{4}.$$

(The numbers 4, 5 and 6 of each colour only matter if the count of balls of a colour could run out, which it cannot here.)