JEE Mathematics 2025 Question with Solution

Given: We need to count $5$-digit numbers $d_1d_2d_3d_4d_5$ formed from the digits $\{0,1,2,3,4,5,6,7\}$ such that the number is greater than $50000$ and $d_1 + d_5 \leq 8$.

Find: The total number of such numbers.

Since the number is greater than $50000$, the first digit can only be $d_1 \in \{5,6,7\}$.

Now apply the condition on the first and last digits:

• If $d_1 = 5$, then $d_5 \leq 3$, so $d_5 \in \{0,1,2,3\}$. This gives $4$ choices.
• If $d_1 = 6$, then $d_5 \leq 2$, so $d_5 \in \{0,1,2\}$. This gives $3$ choices.
• If $d_1 = 7$, then $d_5 \leq 1$, so $d_5 \in \{0,1\}$. This gives $2$ choices.

The middle three digits $d_2, d_3, d_4$ are unrestricted except that each must belong to $\{0,1,2,3,4,5,6,7\}$. Therefore, the number of choices is

Count each case separately:

Adding all cases,

Therefore, the total number of such $5$-digit numbers is $4608$. The correct option is B.