CSAT Year Wise (Temporary Link, This will be removed and no PDFs shall be downloaded here. Please download CSAT subject wise files) / सी सैट

Theme:

CSAT 2023

Q36 2023, (Aptitude, not Probability)

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?

किसी डिब्बे में 14 काली गेंदें, 20 नीली गेंदें, 26 हरी गेंदें, 28 पीली गेंदें, 38 लाल गेंदें और 54 सफ़ेद गेंदें हैं। निम्नलिखित कथनों पर विचार कीजिए:
1. यदि डिब्बे में से यादृच्छिक रूप से कोई सी n गेंदें इस प्रकार निकाली जाएँ कि उनमें कम-से-कम एक रंग का एक पूरा समूह अवश्य हो, तो n की लघुतम संख्या 175 है।
2. यदि डिब्बे में से यादृच्छिक रूप से कोई सी m गेंदें इस प्रकार निकाली जाएँ कि उनमें हर रंग की कम-से-कम एक गेंद अवश्य हो, तो m की लघुतम संख्या 167 है।
उपर्युक्त कथनों में से कौन-सा/से सही है/हैं?

Explanation Please wait..

Statement 1: Value of n: The minimum number of balls needed to draw to ensure at least one full group of a single color is found.

Approach:

Worst-case scenario: Draw maximum number of balls without selecting a full group of any color.

Method: Suppose a colour group has p balls, let’s select (p-1) balls from each group.

13 black + 19 blue + 25 green + 27 yellow + 37 red + 53 white balls. = 174 balls.

Selecting 174 balls is the worst-case scenario, i.e. without forming a full group of any color.

Hence, The value of n, representing the total number of balls needed to guarantee selecting a full group of a certain color, will be 174 + 1 = 175.

So, Statement 1 is correct.

Statement 2: Value of m: The minimum number of balls needed to draw to ensure at least one ball of each color is found.

Approach:

Worst-case scenario: Draw maximum number of balls without selecting any ball of a particular colour.

Method: Suppose a colour group has q balls, let’s select q (all) balls from each group, except the smallest group. As the number of black balls is the least, we can skip it.

20 blue + 26 green + 28 yellow + 38 red + 54 = 166 balls.

Selecting 166 balls is the worst-case scenario, where no black ball will be picked.

Hence, the value of m, representing the total number of balls needed to guarantee at least one ball of each color, will be 166 + 1 = 167

So, Statement 2 is correct.

Statement 1:
Value of n: The minimum number of balls needed to draw to ensure at least
one full group of a single color is found.

Approach:

<ul><li> 
Worst-case
scenario:
Draw maximum number of balls without selecting a full group of any color.</li></ul>

<ul><li> 
Method:
Suppose
a colour group has p balls, let’s select (p-1) balls from each group.</li></ul>

<ul><li> 
13
black + 19 blue + 25 green + 27 yellow + 37 red + 53 white balls. = 174 balls.</li></ul>

<ul><li> 
Selecting
174 balls is the worst-case scenario, i.e. without forming a full group of any
color.</li></ul>

<ul><li> 
Hence,
The value of n, representing the total number of balls needed to guarantee selecting
a full group of a certain color, will be 174 + 1 = 175.</li></ul>

<ul><li> 
So,
Statement 1 is correct.</li></ul>

Statement 2:
Value of m: The minimum number of balls needed to draw to ensure at least
one ball of each color is found.

Approach:

<ul><li> 
Worst-case
scenario:
Draw maximum number of balls without selecting any ball of a particular colour.</li></ul>

<ul><li> 
Method:
Suppose
a colour group has q balls, let’s select q (all) balls from each group, except
the smallest group. As the number of black balls is the least, we can skip it.</li></ul>

<ul><li> 
20
blue + 26 green + 28 yellow + 38 red + 54 = 166 balls.</li></ul>

<ul><li> 
Selecting
166 balls is the worst-case scenario, where no black ball will be picked.</li></ul>

<ul><li> 
Hence,
the value of m, representing the total number of balls needed to guarantee at
least one ball of each color, will be 166 + 1 = 167</li></ul>

<ul><li> 
So,
Statement 2 is correct.</li></ul>

CSAT Year Wise (Temporary Link, This will be removed and no PDFs shall be downloaded here. Please download CSAT subject wise files) / सी सैट

Q36 2023, (Aptitude, not Probability)

Explanation Please wait..

व्याख्या Please wait..

Quick Links

John Doe

CSAT Year Wise (Temporary Link, This will be removed and no PDFs shall be downloaded here. Please download CSAT subject wise files) / सी सैट

Q36 2023, (Aptitude, not Probability)

Explanation Please wait..

व्याख्या Please wait..

Quick Links

Question Meta

Ancient History / प्राचीन इतिहास

John Doe