Maths Q21 – JEE-MAIN 2025

Home Maths Permutations and CombinationsQuestion #326

If the number of seven-digit numbers, such that the sum of their digits is even, is

m \cdot n \cdot 10^{n}

;

m, n \in {1, 2, 3, . . ., 9}

, then

m + n

is equal to _______

Enter Calculated Numerical Value:

Clue 1 of 3: Parity Principle

The sum of digits is even if there is an even number of odd digits. This can be simplified by considering the parity of the sum of the first $k - 1$ digits and the $k$ -th digit.

🧠 Full Solution Path

Step 1: Calculate the number of seven-digit numbers with an even sum of digits✦ Active

A seven-digit number $d_{1} d_{2} d_{3} d_{4} d_{5} d_{6} d_{7}$ has $d_{1} \in {1, . . ., 9}$ (9 choices) and $d_{i} \in {0, . . ., 9}$ for $i = 2, . . ., 7$ (10 choices each). Consider the first six digits $d_{1}, . . ., d_{6}$ . The number of ways to choose these digits is $9 \times 10^{5}$ . Let $S_{6} = d_{1} + . . . + d_{6}$ . The parity of $S_{6}$ can be either even or odd.

For the total sum $S_{7} = S_{6} + d_{7}$ to be even:

- If $S_{6}$ is even, $d_{7}$ must be even. There are 5 even digits (0, 2, 4, 6, 8).

- If $S_{6}$ is odd, $d_{7}$ must be odd. There are 5 odd digits (1, 3, 5, 7, 9).

In both scenarios, for any combination of the first six digits, there are exactly 5 choices for $d_{7}$ that make the total sum $S_{7}$ even. Therefore, the total number of seven-digit numbers with an even sum of digits is $9 \times 10^{5} \times 5 = 45 \times 10^{5}$ .

Step 2: Match the result with the given form

m \cdot n \cdot 10^{n}

○ Expand

We are given that the number of such seven-digit numbers is $m \cdot n \cdot 10^{n}$ , where $m, n \in {1, 2, 3, . . ., 9}$ . We have calculated this number to be $45 \times 10^{5}$ . Comparing the two forms:

m \cdot n \cdot 10^{n} = 45 \times 10^{5}

By comparing the powers of 10, we can deduce $n = 5$ . Substituting $n = 5$ into the equation:

m \cdot 5 = 45

Solving for $m$ , we get $m = \frac{45}{5} = 9$ . Both $m = 9$ and $n = 5$ satisfy the condition $m, n \in {1, 2, . . ., 9}$ .

💡 Teacher's Secret Hint

Ensure that the values of $m$ and $n$ fall within the specified range of ${1, 2, . . ., 9}$ .

Step 3: Calculate

m + n

○ Expand

Now, we need to find the value of $m + n$ :

m + n = 9 + 5 = 14

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Maths Question 21 – JEE-MAIN 2025

Clue 1 of 3: Parity Principle

🧠 Full Solution Path