How many arrangement of the letters in MISSISSIPPI have at least 2 adjacent S's?
I was thinking that I can glue two of the S's together, so there will be 9 letters plus the special letter SS, and the number of arrangements will be $10!/(2!2!4!)$, this is not the correct answer, but what is wrong with my reasoning.
(the correct answer is $11!/(4!4!2!)-7!/(4!2!)\binom{8}{4}$, I understand the solution)
1 Answer
Total arrangements of MISSISSIPPI: $$\frac{11!}{4!4!2!1!}=34650$$
Total arrangements of MIIIPPI: $$\frac{7!}{4!2!1!}=105$$
... and ways to insert SSSS into the $8$ gaps without any adjacent Ss into each variant: $$ {8 \choose 4 }= 70$$
So number of arrangements of MISSISSIPPI that have two Ss together: $$ 34650-105\cdot 70 = 34650-7350 = 27300 $$
Your reasoning neglects the other ways that $2$ Ss can appear together.
