Question

# What is the probability that four Sâ€™s come consecutively in the word â€˜MISSISSIPPIâ€™A. $\dfrac{4}{{165}}$ B.$\dfrac{{16}}{{165}}$ C.$\dfrac{{12}}{{165}}$ D.$\dfrac{5}{{165}}$

Hint: We 1st take count of all 11 letters and rearrange them in total number of outcomes and then take Sâ€™s as a single letter and then find the favorable outcomes.

Total number of letters in MISSISSIPPI = 11 letters (4 â€“ S, 4 â€“ I, 2 â€“ P, 1 â€“ 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 Sâ€™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 â€˜Sâ€™ 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 Sâ€™s which come consecutively if all the letters of the word MISSISSIPPI are rearranged randomly.