JEE Mathematics 2023 Question with Solution

Given: We need four-digit numbers strictly between $5000$ and $10000$, formed from the digits $\{1,3,5,7,9\}$ without repetition.

Find: The total number of such numbers.

To be greater than $5000$, the first digit must be $5$, $7$, or $9$. So, the first place has $3$ choices.

After choosing the first digit, $4$ digits remain for the second place, then $3$ for the third place, and $2$ for the fourth place.

Thus, the total number of numbers is

$$3 \times 4 \times 3 \times 2 = 72$$

Therefore, the number of such numbers is $72$. The correct option is D.

Given: The number must lie between $5000$ and $10000$ and use digits $1,3,5,7,9$ without repetition.

Find: The number of valid arrangements.

A quick way is to count position-wise. The thousand's place must be one of the allowed digits at least $5$, so it has $3$ choices: $5,7,9$. Then arrange any $3$ of the remaining $4$ digits in order:

$${}^4P_3 = 4 \times 3 \times 2$$

Hence,

$$3 \times {}^4P_3 = 3 \times 4 \times 3 \times 2 = 72$$

Therefore, the required count is $72$.