Let X = {x | x = 2 + 4k, where k = 0, 1, 2, 3,...24}. Let S be a subset of X such that the sum of no two elements of S is 100. What is the maximum possible number of elements in S ?  

This question was previously asked in
CDS Elementary Mathematics 16 April 2023 Official Paper
View all CDS Papers >
  1. 10
  2. 11
  3. 12
  4. 13

Answer (Detailed Solution Below)

Option 4 : 13
Free
UPSC CDS 01/2025 General Knowledge Full Mock Test
7.9 K Users
120 Questions 100 Marks 120 Mins

Detailed Solution

Download Solution PDF

Calculation:

The set X is given by

{2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98}.

We want to find the maximum size of a subset S of X such that no two elements sum to 100.

The pairs in X that sum to 100 are

(2, 98), (6, 94), (10, 90), (14, 86), (18, 82), (22, 78), (26, 74), (30, 70), (34, 66), (38, 62), (42, 58), (46, 54), (50, 50){note: 50 appears only once in X }

Therefore, 

To maximize the number of elements in S while ensuring no two elements sum to 100:

  • Choose one element from each of the 12 pairs (but not both)
  • Additionally, include the element 50

 

The maximum possible number of elements in S = 13

∴ The maximum possible number of elements in S be 13.

Latest CDS Updates

Last updated on May 29, 2025

-> The UPSC CDS 2 Notification has been released at upsconline.gov.in. for 453 vacancies.

-> Candidates can apply online from 28th May to 17th June 2025.

-> The CDS 2 Exam will be held on 14th September 2025.

-> Attempt UPSC CDS Free Mock Test to boost your score.

-> The selection process includes Written Examination, SSB Interview, Document Verification, and Medical Examination.  

-> Refer to the CDS Previous Year Papers to enhance your preparation. 

Get Free Access Now
Hot Links: teen patti classic teen patti glory teen patti earning app teen patti gold download