JEE Mathematics 2023 Question with Solution

Given: Digits $3, 5, 6, 7, 8$ are to be used without repetition.

Find: The number of integers greater than $7000$.

Consider two cases.

Case 1: Four digit numbers greater than $7000$

The thousand's digit must be $7$ or $8$, so there are $2$ choices. The remaining three places can be filled by arranging any $3$ of the remaining $4$ digits.

So, the number of four digit numbers greater than $7000$ is $48$.

Case 2: Five digit numbers

Every five digit number formed using all the given digits is automatically greater than $7000$. Hence the number of such numbers is

Therefore, total numbers greater than $7000$ are

So the count is $168$. The solution states the correct value is $168$, although it labels the option as A. Since $168$ corresponds to option B in the given options, the correct option is B.

Given: The digits are $3, 5, 6, 7, 8$ and repetition is not allowed.

Find: How many integers formed are greater than $7000$.

For a number to be greater than $7000$, it can be either a four digit number exceeding $7000$ or any five digit number.

For four digit numbers, the first digit must be greater than or equal to $7$. From the given digits, only $7$ and $8$ are possible.

After choosing the first digit, the remaining three positions are filled from the remaining four digits:

Since there are $2$ choices for the first digit,

For five digit numbers, all $5$ digits are used once each. Therefore, the total number is

Adding both cases,

Therefore, the number of integers is $168$ and the correct option is B.