Answer:
Step-by-step explanation:
The word "LAMEFIREALARM" word consists of
2L , 2M, 2E , 2R , 3A , F and I
no of words with the above letters such that M's are not together
=Total No of words- M's are together
Total no of words![=\frac{13!}{4\times 2!\times 3!}=64,864,800](https://tex.z-dn.net/?f=%3D%5Cfrac%7B13%21%7D%7B4%5Ctimes%202%21%5Ctimes%203%21%7D%3D64%2C864%2C800)
When M's are together . considering 2 M's as one
therefore there are ![\frac{12!}{3\times 2!\times 3!}=9,979,200](https://tex.z-dn.net/?f=%5Cfrac%7B12%21%7D%7B3%5Ctimes%202%21%5Ctimes%203%21%7D%3D9%2C979%2C200)
No of ways![=\frac{13!}{4\times 2!\times 3!}-\frac{12!}{3\times 2!\times 3!}=54,885,600](https://tex.z-dn.net/?f=%3D%5Cfrac%7B13%21%7D%7B4%5Ctimes%202%21%5Ctimes%203%21%7D-%5Cfrac%7B12%21%7D%7B3%5Ctimes%202%21%5Ctimes%203%21%7D%3D54%2C885%2C600)
(b)No of ways such that M's are separated by at least 2 letters
at least 2 letter means 2 letter, 3 letter ......11 letters
So we have to subtract no of ways where there are 1 letter in between M's from total no of ways where 2 M's are not next to each other
No of ways in which there is 1 letter between 2 M's
This can be done by considering 11 cases
In first case Place first M in Starting Position and 2 M on third place
Second case place First M in 2 nd Position and second M on 4 th place
Similarly For 11th case
Place first M in 11th place and second M on 13th place
Total ways
![=\frac{11\times 11!}{3!\times 2!\times 2!\times 2!}](https://tex.z-dn.net/?f=%20%3D%5Cfrac%7B11%5Ctimes%2011%21%7D%7B3%21%5Ctimes%202%21%5Ctimes%202%21%5Ctimes%202%21%7D)
So , the total number of arrangements of the letters of the word LAMEFIREALARM where the two M's are separated by atleast two letters is
![=\frac{13!}{4\times 2!\times 3!}-\frac{12!}{3\times 2!\times 3!}-\frac{11\times 11!}{3!\times 2!\times 2!\times 2!}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B13%21%7D%7B4%5Ctimes%202%21%5Ctimes%203%21%7D-%5Cfrac%7B12%21%7D%7B3%5Ctimes%202%21%5Ctimes%203%21%7D-%5Cfrac%7B11%5Ctimes%2011%21%7D%7B3%21%5Ctimes%202%21%5Ctimes%202%21%5Ctimes%202%21%7D)
![=45738000](https://tex.z-dn.net/?f=%3D45738000)