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Find the number of 4x4 arrays whose elements belong to { 0, 1, 2, 3). This is also mentioned that sum of the numbers in each row & each column is divisible by 4. (4x4 array means arrangement of 16 elements arranged in 4 rows & 4 columns.)?
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Find the number of 4x4 arrays whose elements belong to { 0, 1, 2, 3). ...
Introduction
To find the number of 4x4 arrays where the elements belong to {0, 1, 2, 3} and the sum of the numbers in each row and each column is divisible by 4, we utilize combinatorial counting with modular arithmetic.
Step 1: Understanding Modulo Constraints
- Each element can be represented in modulo 4, meaning the sums of the elements in each row and column must satisfy:
- \( S_{row_i} \equiv 0 \mod 4 \) for all rows \( i \) (where \( i = 1, 2, 3, 4 \))
- \( S_{column_j} \equiv 0 \mod 4 \) for all columns \( j \) (where \( j = 1, 2, 3, 4 \))
Step 2: Counting Arrays
- Each row can have combinations of elements such that their total sum is divisible by 4. We will use the concept of generating functions to count these valid combinations.
- The generating function for each element can be expressed as:
- \( f(x) = 1 + x + x^2 + x^3 \)
- The polynomial representation \( f(x)^4 \) (for 4 elements in each row) is evaluated to find coefficients corresponding to terms where \( x^{4k} \) (for \( k \in \mathbb{Z} \)) reflects sums divisible by 4.
Step 3: Applying the Inclusion-Exclusion Principle
- To satisfy the conditions for both rows and columns simultaneously, we can apply the inclusion-exclusion principle to account for overlaps in valid configurations.
Conclusion
- The total number of valid 4x4 arrays can be computed using advanced combinatorial techniques, leading to the final count based on the principles outlined.
- This approach ensures that both row and column conditions are satisfied, yielding a specific count of arrays adhering to the given constraints.
The final result can be derived from the calculations based on the above methodologies.
To find the number of 4x4 arrays where the elements belong to {0, 1, 2, 3} and the sum of the numbers in each row and each column is divisible by 4, we utilize combinatorial counting with modular arithmetic.
Step 1: Understanding Modulo Constraints
- Each element can be represented in modulo 4, meaning the sums of the elements in each row and column must satisfy:
- \( S_{row_i} \equiv 0 \mod 4 \) for all rows \( i \) (where \( i = 1, 2, 3, 4 \))
- \( S_{column_j} \equiv 0 \mod 4 \) for all columns \( j \) (where \( j = 1, 2, 3, 4 \))
Step 2: Counting Arrays
- Each row can have combinations of elements such that their total sum is divisible by 4. We will use the concept of generating functions to count these valid combinations.
- The generating function for each element can be expressed as:
- \( f(x) = 1 + x + x^2 + x^3 \)
- The polynomial representation \( f(x)^4 \) (for 4 elements in each row) is evaluated to find coefficients corresponding to terms where \( x^{4k} \) (for \( k \in \mathbb{Z} \)) reflects sums divisible by 4.
Step 3: Applying the Inclusion-Exclusion Principle
- To satisfy the conditions for both rows and columns simultaneously, we can apply the inclusion-exclusion principle to account for overlaps in valid configurations.
Conclusion
- The total number of valid 4x4 arrays can be computed using advanced combinatorial techniques, leading to the final count based on the principles outlined.
- This approach ensures that both row and column conditions are satisfied, yielding a specific count of arrays adhering to the given constraints.
The final result can be derived from the calculations based on the above methodologies.
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Find the number of 4x4 arrays whose elements belong to { 0, 1, 2, 3). This is also mentioned that sum of the numbers in each row & each column is divisible by 4. (4x4 array means arrangement of 16 elements arranged in 4 rows & 4 columns.)?
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Find the number of 4x4 arrays whose elements belong to { 0, 1, 2, 3). This is also mentioned that sum of the numbers in each row & each column is divisible by 4. (4x4 array means arrangement of 16 elements arranged in 4 rows & 4 columns.)? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Find the number of 4x4 arrays whose elements belong to { 0, 1, 2, 3). This is also mentioned that sum of the numbers in each row & each column is divisible by 4. (4x4 array means arrangement of 16 elements arranged in 4 rows & 4 columns.)? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the number of 4x4 arrays whose elements belong to { 0, 1, 2, 3). This is also mentioned that sum of the numbers in each row & each column is divisible by 4. (4x4 array means arrangement of 16 elements arranged in 4 rows & 4 columns.)?.
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