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What is the probability that four S’s come consecutively in the word ‘MISSISSIPPI’
A. $\dfrac{4}{{165}}$
B.$\dfrac{{16}}{{165}}$
C.$\dfrac{{12}}{{165}}$
D.$\dfrac{5}{{165}}$
Answer
![Verified]()
Verified
Verified
Hint: We 1st take count of all 11 letters and rearrange them in total number of outcomes and then take S’s as a single letter and then find the favorable outcomes.
Complete step-by-step answer:
Total number of letters in MISSISSIPPI = 11 letters (4 – S, 4 – I, 2 – P, 1 – M)
Total number of ways of arranging MISSISSIPPI $ = \dfrac{{11!}}{{\left( {4!} \right)\left( {4!} \right)\left( {2!} \right)}}$
Now we need all S to be together
So, if we consider SSSS as 1 block then the remaining number of letters will be MIIIPPI = 8 letters
Number of ways of such arrangement $ = \dfrac{{8!}}{{\left( {4!} \right)\left( {2!} \right)}} = \dfrac{{8!}}{{4! \times 2}}$
Probability that all 4 S’s are together $ = \dfrac{2}{1}$
=$ = \dfrac{{8!}}{{\left( {4!} \right) \times 2}} \times = \dfrac{{\left( {4!} \right)\left( {4!} \right)\left( {2!} \right)}}{{11!}}$
$ = \dfrac{{8!}}{{11 \times 10 \times 9 \times 8!}} \times 4!$
$ = \dfrac{{4 \times 3 \times 2 \times 1}}{{11 \times 10 \times 9}} = \dfrac{4}{{165}}$
Therefore the probability of all ‘S’ are together =$\dfrac{4}{{165}}$
The correct answer is option (A)
Note: To solve such a question we first see what is the probability that four S’s which come consecutively if all the letters of the word MISSISSIPPI are rearranged randomly.
Complete step-by-step answer:
Total number of letters in MISSISSIPPI = 11 letters (4 – S, 4 – I, 2 – P, 1 – M)
Total number of ways of arranging MISSISSIPPI $ = \dfrac{{11!}}{{\left( {4!} \right)\left( {4!} \right)\left( {2!} \right)}}$
Now we need all S to be together
So, if we consider SSSS as 1 block then the remaining number of letters will be MIIIPPI = 8 letters
Number of ways of such arrangement $ = \dfrac{{8!}}{{\left( {4!} \right)\left( {2!} \right)}} = \dfrac{{8!}}{{4! \times 2}}$
Probability that all 4 S’s are together $ = \dfrac{2}{1}$
=$ = \dfrac{{8!}}{{\left( {4!} \right) \times 2}} \times = \dfrac{{\left( {4!} \right)\left( {4!} \right)\left( {2!} \right)}}{{11!}}$
$ = \dfrac{{8!}}{{11 \times 10 \times 9 \times 8!}} \times 4!$
$ = \dfrac{{4 \times 3 \times 2 \times 1}}{{11 \times 10 \times 9}} = \dfrac{4}{{165}}$
Therefore the probability of all ‘S’ are together =$\dfrac{4}{{165}}$
The correct answer is option (A)
Note: To solve such a question we first see what is the probability that four S’s which come consecutively if all the letters of the word MISSISSIPPI are rearranged randomly.
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