<p>The number of ways to fill each of the four cells of the table with a distinct natural number such that the sum of the numbers is 10 and the sums of the numbers placed diagonally are equal is</p>
Step-by-Step Solution
Key Concept: Set up equations where diagonal sums are equal: if cells are arranged as a 2×2 grid, then a+d = b+c. Combined with the constraint a+b+c+d = 10, you can reduce variables and count valid assignments of distinct natural numbers.
<p><strong>Step 1: Set up the constraint equations.</strong></p><p>Let the 2×2 table be:</p><p>⎡ a b ⎤</p><p>⎣ c d ⎦</p><p>Given: a + b + c + d = 10 and a + d = b + c (diagonal equality)</p><p><strong>Step 2: Simplify using diagonal constraint.</strong></p><p>From a + d = b + c, we get: a + d = b + c</p><p>Substituting into the sum: (a + d) + (b + c) = 10</p><p>Since a + d = b + c, let a + d = b + c = k</p><p>Then 2k = 10, so k = 5</p><p><strong>Step 3: Find pairs of distinct natural numbers summing to 5.</strong></p><p>Pairs {x, y} where x + y = 5 and x ≠ y:</p><p>• {1, 4}: 1 + 4 = 5 ✓</p><p>• {2, 3}: 2 + 3 = 5 ✓</p><p><strong>Step 4: Assign pairs to diagonals.</strong></p><p>We need (a, d) to be one of {1,4}, {4,1}, {2,3}, {3,2} and (b, c) to be another valid pair from remaining numbers.</p><p>If {a,d} = {1,4}: then {b,c} must use {2,3}</p><p>• (a,d) = (1,4), (b,c) = (2,3): ✓</p><p>• (a,d) = (1,4), (b,c) = (3,2): ✓</p><p>• (a,d) = (4,1), (b,c) = (2,3): ✓</p><p>• (a,d) = (4,1), (b,c) = (3,2): ✓</p><p>If {a,d} = {2,3}: then {b,c} must use {1,4}</p><p>• (a,d) = (2,3), (b,c) = (1,4): ✓</p><p>• (a,d) = (2,3), (b,c) = (4,1): ✓</p><p>• (a,d) = (3,2), (b,c) = (1,4): ✓</p><p>• (a,d) = (3,2), (b,c) = (4,1): ✓</p><p><strong>Step 5: Count total arrangements.</strong></p><p>Total = 8 ways</p><p>∴ Answer: A</p>
Correct Answer: A