Question
In how many ways can the letters of the word “MISSISSIPPI” be arranged? Of these arrangements, how many have all four S’s together? JEE Main 2024 pattern.
Solution — Step by Step
Total letters: 11. Frequencies: M = 1, I = 4, S = 4, P = 2.
Total arrangements with repetition adjustments:
Compute: . .
Glue the four S’s into one super-letter. Now we have 8 entities to arrange: M, I, I, I, I, P, P, and the SSSS block.
Arrangements of 8 entities with repetitions:
Total arrangements: . Arrangements with all S’s together: .
Why This Works
The formula counts permutations of objects when there are groups of identical objects. We divide by the factorials of group sizes because permuting identical objects within a group doesn’t produce a new arrangement.
The “all together” trick is universal: glue them into a single entity, count arrangements, and (if those entities are distinguishable internally) multiply by the internal arrangements. Here all four S’s are identical, so internal arrangements contribute a factor of 1.
Alternative Method
For “S’s together”, you could use the gap method: arrange the other 7 letters (M, I, I, I, I, P, P) in ways, then place the SSSS block in any of the 8 gaps. Total: . Same answer.
Students often forget to divide by the factorial of the count of each repeated letter. With four I’s, four S’s, and two P’s, you must divide by , not just .
For “always together” problems: glue the special letters into a block, then count arrangements of the new (smaller) set. For “never together” problems: use total arrangements minus “together” arrangements, or use the gap method.