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The first n natural numbers, 1 to n, have to be arranged in a row from left to right. The n numbers are arranged such that there are an odd number of numbers between any two even numbers as well as between any two odd numbers. If the number of ways in which this can be done is 72, then find the value of n.
(2016)
- a)6
- b)7
- c)8
- d)More than 8
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The first n natural numbers, 1 to n, have to be arranged in a row from...
If n is even, i.e., say n = 2 m then the number of ways is 2 × m! × m!, i.e., m odd numbers in alternate places and m even numbers in alternate places.
If n is odd, i.e., say n = 2m + 1, then the number of ways = m!(m + 1)!
Hence, either 2(m!)2 = 72 or m! (m + 1)! = 72 If 2(m!)2 = 72 ⇒ m! = 6 ⇒ m = 3
for m!(m + 1)! = 72, there is no solution.
Hence m = 3, and n = 2m = 6.
If n is odd, i.e., say n = 2m + 1, then the number of ways = m!(m + 1)!
Hence, either 2(m!)2 = 72 or m! (m + 1)! = 72 If 2(m!)2 = 72 ⇒ m! = 6 ⇒ m = 3
for m!(m + 1)! = 72, there is no solution.
Hence m = 3, and n = 2m = 6.
Most Upvoted Answer
The first n natural numbers, 1 to n, have to be arranged in a row from...
Problem Analysis:
In this problem, we are given that the first n natural numbers are arranged in a row from left to right. We need to find the value of n such that there are an odd number of numbers between any two even numbers as well as between any two odd numbers.
Understanding the Constraints:
Let's try to understand the constraints mentioned in the problem:
1. There should be an odd number of numbers between any two even numbers: This means that if we have two even numbers, say a and b, there should be an odd number of numbers between them. For example, if a = 2 and b = 4, then there should be 1 number between them (i.e., 3). Similarly, if a = 6 and b = 10, there should be 2 numbers between them (i.e., 7 and 9).
2. There should be an odd number of numbers between any two odd numbers: This means that if we have two odd numbers, say c and d, there should be an odd number of numbers between them. For example, if c = 3 and d = 5, then there should be 1 number between them (i.e., 4). Similarly, if c = 7 and d = 11, there should be 2 numbers between them (i.e., 9 and 10).
Solution Approach:
Let's try to solve this problem by considering different values of n and analyzing the arrangements:
1. n = 1: In this case, there is only one number (i.e., 1). The given condition is satisfied as there are no two even or odd numbers to check the condition. Hence, there is only one possible arrangement.
2. n = 2: In this case, there are two numbers (i.e., 1 and 2). The given condition is satisfied as there is only one odd number (i.e., 1) and one even number (i.e., 2). Hence, there is only one possible arrangement.
3. n = 3: In this case, there are three numbers (i.e., 1, 2, and 3). The given condition is not satisfied as there are two odd numbers (i.e., 1 and 3) and one even number (i.e., 2). Hence, it is not possible to arrange the numbers in a way that satisfies the given condition.
4. n = 4: In this case, there are four numbers (i.e., 1, 2, 3, and 4). The given condition is satisfied as there is one odd number (i.e., 1), one even number (i.e., 2), one odd number (i.e., 3), and one even number (i.e., 4). Hence, there is only one possible arrangement.
5. n = 5: In this case, there are five numbers (i.e., 1, 2, 3, 4, and 5). The given condition is not satisfied as there are two odd numbers (i.e., 1 and 3) and two even numbers (i.e., 2 and 4). Hence, it is not possible to arrange the numbers in a way that satisfies the given condition.
6. n = 6: In this case,
In this problem, we are given that the first n natural numbers are arranged in a row from left to right. We need to find the value of n such that there are an odd number of numbers between any two even numbers as well as between any two odd numbers.
Understanding the Constraints:
Let's try to understand the constraints mentioned in the problem:
1. There should be an odd number of numbers between any two even numbers: This means that if we have two even numbers, say a and b, there should be an odd number of numbers between them. For example, if a = 2 and b = 4, then there should be 1 number between them (i.e., 3). Similarly, if a = 6 and b = 10, there should be 2 numbers between them (i.e., 7 and 9).
2. There should be an odd number of numbers between any two odd numbers: This means that if we have two odd numbers, say c and d, there should be an odd number of numbers between them. For example, if c = 3 and d = 5, then there should be 1 number between them (i.e., 4). Similarly, if c = 7 and d = 11, there should be 2 numbers between them (i.e., 9 and 10).
Solution Approach:
Let's try to solve this problem by considering different values of n and analyzing the arrangements:
1. n = 1: In this case, there is only one number (i.e., 1). The given condition is satisfied as there are no two even or odd numbers to check the condition. Hence, there is only one possible arrangement.
2. n = 2: In this case, there are two numbers (i.e., 1 and 2). The given condition is satisfied as there is only one odd number (i.e., 1) and one even number (i.e., 2). Hence, there is only one possible arrangement.
3. n = 3: In this case, there are three numbers (i.e., 1, 2, and 3). The given condition is not satisfied as there are two odd numbers (i.e., 1 and 3) and one even number (i.e., 2). Hence, it is not possible to arrange the numbers in a way that satisfies the given condition.
4. n = 4: In this case, there are four numbers (i.e., 1, 2, 3, and 4). The given condition is satisfied as there is one odd number (i.e., 1), one even number (i.e., 2), one odd number (i.e., 3), and one even number (i.e., 4). Hence, there is only one possible arrangement.
5. n = 5: In this case, there are five numbers (i.e., 1, 2, 3, 4, and 5). The given condition is not satisfied as there are two odd numbers (i.e., 1 and 3) and two even numbers (i.e., 2 and 4). Hence, it is not possible to arrange the numbers in a way that satisfies the given condition.
6. n = 6: In this case,
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The first n natural numbers, 1 to n, have to be arranged in a row from left to right. The n numbers are arranged such that there are an odd number of numbers between any two even numbers as well as between any two odd numbers. If the number of ways in which this can be done is 72, then find the value of n.(2016)a)6b)7c)8d)More than 8Correct answer is option 'A'. Can you explain this answer?
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