JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Permutations and Combinations

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

Answer: 14

Solution

A seven-digit number has a leading digit from

1

9

and six further digits from

0

9

, giving

9 \times 1 0^{6}

numbers. For any fixed first six digits, the last digit can be completed in exactly five ways to make the digit sum even and five ways to make it odd, so precisely half the numbers have an even digit sum:

\frac{9 \times 1 0 ^{6}}{2} = 4500000 = 9 \cdot 5 \cdot 1 0^{5}

. Matching

m \cdot n \cdot 1 0^{n}

gives

m = 9, n = 5

, so

m + n = 14

JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Integer answer

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