JAMIA 2023 — Mathematics PYQ

Counting

1. Words starting with 'A':

Remaining letters: {E, H, M, S} (4 letters)

Number of words = $4!=24$

(Total words so far: $24$)

2. Words starting with 'E':

Remaining letters: {A, H, M, S}

Number of words = $4!=24$

(Total words so far: $24+24=48$)

3. Words starting with 'H':

Remaining letters: {A, E, M, S}

Number of words = $4!=24$

(Total words so far: $48+24=72$)

4. Words starting with 'M':

We need to reach the 86th word. Currently, we are at 72. We need $86-72=14$ more words. Let's look at words starting with MA:

• MA... (Remaining: {E, H, S}) $\to 3!=6$ words. (Total: $72+6=78$)

• ME... (Remaining: {A, H, S}) $\to 3!=6$ words. (Total: $78+6=84$)

5. Words starting with 'MH':

We are at 84 words. The 85th and 86th words will start with MH.

The remaining letters are {A, E, S}. In alphabetical order, they are: A, E, S.

• 85th word: M H A E S

• 86th word: M H A S E

Final Result

The 86th word is MHASE.

The last letter of the 86th word is E.