GATE Civil 2023 Set 2 | GA | Question: 4

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. $2$

Answer 1

The Pigeonhole Principle tells us that if we have more objects to put into boxes than there are boxes, at least one box must contain multiple objects.

In this case, each color (red, green, and blue) can be considered a "box," and the balls we pick are the objects. We want to find the smallest value of $N$ such that we are guaranteed to have at least two balls of the same color.

The worst-case scenario for this problem is when we pick one ball of each color $\text{(1 red, 1 green, 1 blue)}.$ In this case, $N$ would be $3,$ and we still don't have two balls of the same color. However, if we pick just one more ball (regardless of its color), we will have at least two balls of the same color.

So, the smallest value of $N$ that guarantees there will be at least two balls of the same color is $4.$

Correct Answer: A

References:

• https://www.eecs.yorku.ca/course_archive/2008-09/S/1019/Website_files/19-the-pigeonhole-principle.pdf
• https://math.berkeley.edu/~rhzhao/10BSpring19/Worksheets/Discussion%203%20Solutions.pdf
• https://brilliant.org/wiki/pigeonhole-principle-definition/

Comments

according to question if we cannot see the ball while drawing then will there be possibility that we can draw two same color balls in first two draw?

 

 

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Bhai in the worst case we could have Min 4 Balls and Usmien se 2 Same color ke nikal jaaye
yehh counting waale Problems worst case mein check karneka

Answer 2

If we pick 4 balls from the box, even if the worst case has 3 different colors, 4th one will match any color. So answer is 4.

Answer 3

Colours available
     ┌────────┬────────┬────────┐
     │  Red   │ Green  │  Blue  │
     └────────┴────────┴────────┘
          1        1        1
          │        │        │
          └────────┴────────┴───┐
                                │
                     First 3 picks:
                   All different colours
                                │
                                ▼
                       Pick 4th ball
                                │
                                ▼
              Must be Red / Green / Blue
                                │
                                ▼
            At least 2 balls of same colour