How many distinct permutations of the letters in Mississippi do the four not come together?

In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together?

Answer

Hint: It is evident from the question that we need to find the permutations for the word MISSISSIPPI by applying the given conditions. In the given word we can observe that the letters of the word are repeating like S, I and P. The number of permutations of n objects with \[{{n}_{1}}\] identical objects of type 1,\[{{n}_{2}}\] identical objects of type 2,…….., and \[{{n}_{k}}\] identical objects of type k is \[\dfrac{n!}{{{n}_{1}}!{{n}_{2}}!.......{{n}_{k}}!}\] .

Complete step-by-step answer:
By observing the question clearly, it has been asked to find the distinct permutations of the letters in MISSISSIPPI when the four I’s ‘do not’ come together
It can be interpreted that
Total number of permutations of four I’s not coming together=total number of permutations – total number of permutations with the four I’s coming together.
Firstly, let us find the total number of permutations in the word MISSISSIPPI
The word MISSISSIPPI has four S’s, four I’s, two P’s and one M
It can be seen that the letters of the word are repeating so the formula \[\dfrac{n!}{{{n}_{1}}!{{n}_{2}}!.......{{n}_{k}}!}\] can be used to find the total number of permutations
Total number of letters in the word =\[4+4+2+1=11\]
Hence,
\[\begin{align}
  & n=11 \\
 & k=3 \\
 & {{n}_{1}}=4 \\
 & {{n}_{2}}=4 \\
 & {{n}_{3}}=2 \\
\end{align}\]
Substituting the values in the above formula we get
Total number of permutations=\[\dfrac{11!}{4!4!2!}=\dfrac{39916800}{24\times 24\times 2}=34650\]
Secondly now let us find the number of permutations of four I’s together which means in any arrangement of the given word the four I’s must be together
For that we can consider the four I’s as a single object
MISSISSIPPI 🡪MSSSSPP[IIII]
Now again it’s just like the first part here we have four S’s and two P’s repeating
So here the total number of letters=four S’s + two P’s + one M+ one object (four I’s together)
\[\begin{align}
  & n=8 \\
 & k=2 \\
 & {{n}_{1}}=4 \\
 & {{n}_{2}}=2 \\
\end{align}\]
Substituting the values in the given formula \[\dfrac{n!}{{{n}_{1}}!{{n}_{2}}!.......{{n}_{k}}!}\] we get
Total number of permutations with four I’s together=\[\dfrac{8!}{4!2!}=\dfrac{40320}{24\times 2}=840\]
Total number of permutations of 4I’s not coming together=total number of permutations – total number of permutations with the 4I’s coming together.
Total number of permutations of 4I’s not coming together=\[34650-840=33810\]
Hence in 33810 distinct permutations of the letters in MISSISSIPPI the four I’s do not come together.

Note: If the student starts to solve the problem by finding the total number of permutations of 4I’s not coming together it would be so big and lengthy to solve. There would always be an easy way to solve the problem and it could be conquered by thinking a bit more logically. While considering the four I’s together it should be considered as a single object rather than four I’s this point should be noted carefully for an accurate answer.

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