A box contains 14 black balls, 20 blue balls, 26 green balls, 28 yellow balls, 38 red balls and 54 white balls. Consider the following statements: 1. The smallest number n such that any n balls drawn from the box randomly must contain one full group of at least one colour is $175$. 2. The smallest number m such that any m balls drawn from the box randomly must contain at least one ball of each colour is $167$. Which of the above statements is/are correct?

QUESTION

CSAT

Easy

Maths

Prelims 2023

A box contains 14 black balls, 20 blue balls, 26 green balls, 28 yellow balls, 38 red balls and 54 white balls.

Consider the following statements:

The smallest number n such that any n balls drawn from the box randomly must contain one full group of at least one colour is $175$ .
The smallest number m such that any m balls drawn from the box randomly must contain at least one ball of each colour is $167$ .

Which of the above statements is/are correct?

Select an option to attempt

Explanation

The box contains 14 black balls, 20 blue balls, 26 green balls, 28 yellow balls, 38 red balls and 54 white balls.

Value of n:

We have to find out the minimum possible number of balls that should be drawn from the box such that the balls drawn must contain one full group of at least one colour. Say, it may have all 14 black balls, or all 20 blue balls, etc.

Let’s think about the worst-case scenario. What is the maximum number of balls that we can draw without selecting a full group of any colour?

Let’s select 13 black balls, 19 blue balls, 25 green balls, 27 yellow balls, 37 red balls and 53 white balls. These are 174 balls in total.

Now, if we select even one more ball (of any colour), it’s a certainty that at least one full group of a certain colour will get selected. So, the value of n = $174 + 1 = 175$ .

So, Statement 1 is correct.

Value of m

We have to find out the minimum possible number of balls that should be drawn from the box such that the balls drawn must contain at least one ball of each colour.

Let’s think about the worst-case scenario. What is the maximum number of balls that we can draw without selecting any ball of a particular colour?

As the number of black balls is the least, we can draw the maximum possible number of balls without selecting a black ball. So, let’s select 20 blue balls, 26 green balls, 28 yellow balls, 38 red balls and 54 white balls. These are 166 balls in total.

Now, only black balls are left. So, the next ball we choose will certainly be a black ball, and we will end up having at least one ball of each colour. So, the value of m = $166 + 1 = 167$