JEE Main 2025 Apr 3 Shift 1, Mathematics Q24: Combinations

JEE Main2025Apr 3, Shift 1Mathematics

If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 1 0^{n}$ ; $m, n \in {1, 2, 3, \dots, 9}$ , then $m + n$ is equal to _____

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Solution

Total seven-digit numbers: the leading digit can be 1–9 (9 choices) and the other six digits 0–9 (10 choices each). Total $= 9 \times 1 0^{6} = 9000000.$

Fix the first six digits freely; the last digit's parity is then forced to make the total even, leaving exactly 5 choices (digits of the chosen parity from {0,...,9}). So exactly half of all seven-digit numbers have an even digit sum:

$\frac{9 \times 1 0 ^{6}}{2} = 9 \times 5 \times 1 0^{5} = 45 \times 1 0^{5} .$

Matching $m \cdot n \cdot 1 0^{n}$ : take $m = 9, n = 5.$

$m + n = 14.$

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JEE Main 2025 Apr 3 Shift 1, Mathematics Q24: Combinations

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