In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together?
Answer
Verified
Hint: In this question we need to find the number of distinct permutations of the word MISSISSIPPI where four I’s not come together. In order to do this, we will find the total permutation and subtract it from total permutation of I coming together. This will help us simplify the question and reach the answer.
Complete step-by-step answer:
We have to find the number of distinct permutations of the word MISSISSIPPI where four I's do not come together.
So, Total number of permutation of 4I not coming together = Total permutation – Total permutation of I’s coming together.
For total permutations in MISSISSIPPI,
As there are repeating characters in the word, so we will use the formula, $\dfrac{{n!}}{{p!q!r!...}}$
We are having 4I’s, 4S’s, 2P’s and 1M
So, the total number of permutations $ = \dfrac{{11!}}{{4!4!2!}} = 34560$.
Now, for total permutations of I’s coming together in MISSISSIPPI,
We will take 4I’s as 1I,
So, the total permutations of I’s coming together $ = \dfrac{{8!}}{{4!2!}} = 840$
As, the total number of permutation of 4I not coming together = Total permutation – Total permutation of I’s coming together
Therefore, the total permutation of 4I’s not coming together = 34650 – 840 = 33810.
Note: Whenever we face such types of problems the value point to remember is that we need to have a good grasp over permutations and its formulas. The basic formula to calculate permutations has been discussed above and used to solve the given question. We must also remember that the approach used above is the best way to solve these types of questions.
In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together?
Answer
Verified
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.