JEE Mathematics 2023 Question with Solution

Given: A $4$-digit ATM pin code has all digits different, the greatest digit is $7$, and if the code is $abcd$, then

Find: The maximum number of trials needed, which equals the total number of valid codes.

Since the greatest digit is $7$ and all digits are different, the four digits are chosen from

with $7$ definitely included.

Let the code be $abcd$. Then

So the total sum becomes

Hence the sum of all four digits must be even.

Now count the $4$-element digit sets containing $7$ that can be split into two pairs having equal sum. The valid sets listed in the solution are

Thus, there are $6$ such sets.

For each valid set of $4$ distinct digits, the number of permutations satisfying

is $12$.

Therefore, total possible codes are

So the maximum number of trials necessary is $72$.

Given: Distinct digits, maximum digit $7$, and equal pair-sum condition.

Find: Total number of valid codes.

A quick way is to first identify all $4$-digit sets containing $7$ for which the digits can be grouped into two pairs with the same sum. The extracted working gives exactly $6$ such sets.

For each such set, once two equal-sum pairs are available, the digits can be arranged into positions $a,b,c,d$ in $12$ valid ways while preserving

Hence the required number is

Therefore, the correct numerical answer is $72$.