JEE Mathematics 2023 Question with Solution

Given: The word MONDAY has $6$ distinct letters.

Find: The dictionary rank of the word MONDAY.

Arrange the letters in alphabetical order:

$$D < A < M < N < O < Y$$

Now count how many words come before MONDAY.

For the first letter M, smaller available letters are A and D. Each such choice gives

$$5! = 120$$

words. So the count is

$$2 \times 120 = 240$$

Fix M. For the second letter O, among the remaining letters A, D, N, O, Y, smaller letters are A, D, N. Each gives

$$4! = 24$$

words. So the count added is

$$3 \times 24 = 72$$

Fix MO. For the third letter N, among A, D, N, Y, smaller letters are A and D. Each gives

$$3! = 6$$

words. So the count added is

$$2 \times 6 = 12$$

Fix MON. For the fourth letter D, among A, D, Y, smaller available letter is A. This gives

$$2! = 2$$

words.

Fix MOND. For the fifth letter A, among A, Y, there is no smaller available letter, so the count added is $0$.

Hence total number of words before MONDAY is

$$240 + 72 + 12 + 2 = 326$$

Therefore, the rank of MONDAY is

$$326 + 1 = 327$$

So, the correct option is B.

The solution states option C, but its computed value is $327$. Since $327$ corresponds to option B in the given options, the correct answer is B.