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Let S = {1, 2, 3,. n} be a set of N natural numbers. Let T be a subset of S such that the sum of any three elements of T is not less than N. Find the maximum number of elements in any such subset T for N = 40? (a) 26 (b) 27 (c) 28 (d) None of these Correct answer is given (d) Can someone explain?
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Let S = {1, 2, 3,. n} be a set of N natural numbers. Let T be a subset...
To find the maximum number of elements in the subset T, we need to consider the worst-case scenario where the sum of any three elements of T is as close to N as possible without being less than N.
Let's assume that the maximum number of elements in T is x.
The sum of any three elements in T can be written as (x-2) + (x-1) + x = 3x - 3. Since the sum of any three elements of T is not less than N, we have the inequality:
3x - 3 ≥ N
To find the maximum value of x, we need to solve this inequality for N = 40.
3x - 3 ≥ 40
3x ≥ 43
x ≥ 43/3
We need to find the largest integer value of x that satisfies this inequality. The largest integer value less than or equal to 43/3 is 14. Therefore, x can be at most 14.
Now, let's consider a subset T with 14 elements: T = {27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}. In this subset, the sum of any three elements is greater than or equal to 40.
If we try to add an additional element to T, we will have to remove an element from T to maintain the maximum number of elements. However, removing an element will decrease the sum of the elements in T, which means that the sum of any three elements of T may be less than N.
Therefore, the maximum number of elements in subset T for N = 40 is 14.
Hence, the correct answer is (d) None of these.
Let's assume that the maximum number of elements in T is x.
The sum of any three elements in T can be written as (x-2) + (x-1) + x = 3x - 3. Since the sum of any three elements of T is not less than N, we have the inequality:
3x - 3 ≥ N
To find the maximum value of x, we need to solve this inequality for N = 40.
3x - 3 ≥ 40
3x ≥ 43
x ≥ 43/3
We need to find the largest integer value of x that satisfies this inequality. The largest integer value less than or equal to 43/3 is 14. Therefore, x can be at most 14.
Now, let's consider a subset T with 14 elements: T = {27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}. In this subset, the sum of any three elements is greater than or equal to 40.
If we try to add an additional element to T, we will have to remove an element from T to maintain the maximum number of elements. However, removing an element will decrease the sum of the elements in T, which means that the sum of any three elements of T may be less than N.
Therefore, the maximum number of elements in subset T for N = 40 is 14.
Hence, the correct answer is (d) None of these.
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Let S = {1, 2, 3,. n} be a set of N natural numbers. Let T be a subset of S such that the sum of any three elements of T is not less than N. Find the maximum number of elements in any such subset T for N = 40? (a) 26 (b) 27 (c) 28 (d) None of these Correct answer is given (d) Can someone explain?
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Let S = {1, 2, 3,. n} be a set of N natural numbers. Let T be a subset of S such that the sum of any three elements of T is not less than N. Find the maximum number of elements in any such subset T for N = 40? (a) 26 (b) 27 (c) 28 (d) None of these Correct answer is given (d) Can someone explain? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Let S = {1, 2, 3,. n} be a set of N natural numbers. Let T be a subset of S such that the sum of any three elements of T is not less than N. Find the maximum number of elements in any such subset T for N = 40? (a) 26 (b) 27 (c) 28 (d) None of these Correct answer is given (d) Can someone explain? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let S = {1, 2, 3,. n} be a set of N natural numbers. Let T be a subset of S such that the sum of any three elements of T is not less than N. Find the maximum number of elements in any such subset T for N = 40? (a) 26 (b) 27 (c) 28 (d) None of these Correct answer is given (d) Can someone explain?.
Let S = {1, 2, 3,. n} be a set of N natural numbers. Let T be a subset of S such that the sum of any three elements of T is not less than N. Find the maximum number of elements in any such subset T for N = 40? (a) 26 (b) 27 (c) 28 (d) None of these Correct answer is given (d) Can someone explain? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Let S = {1, 2, 3,. n} be a set of N natural numbers. Let T be a subset of S such that the sum of any three elements of T is not less than N. Find the maximum number of elements in any such subset T for N = 40? (a) 26 (b) 27 (c) 28 (d) None of these Correct answer is given (d) Can someone explain? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let S = {1, 2, 3,. n} be a set of N natural numbers. Let T be a subset of S such that the sum of any three elements of T is not less than N. Find the maximum number of elements in any such subset T for N = 40? (a) 26 (b) 27 (c) 28 (d) None of these Correct answer is given (d) Can someone explain?.
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