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Hint: Count the total number of letters and the number of times each letter is occurring in the given word and then apply the concept of permutation to find the total number of words that can be formed using the given letters. After having the total number of outcomes, we need to find the favorable number of outcomes, for which we fix the given letters, which is 4S’s coming consecutively, and fit in the others to fill the other blanks and form another table. Now, again apply the concept of permutation on this table to find the favorable number of outcomes. Towards the end, divide the favorable outcomes with the total outcomes to find the probability. After getting the probability, compare it with the given probability to find the value of k.

Let us start by considering the given word ‘MISSISSIPPI’. Let us count the number of each letter occurring in the given word.

Now, we will use the concept of permutation to find out the maximum number of words that can be made with the letters in ‘MISSISSIPPI’. This number would be in turn our maximum number of possible outcomes.

Now, let us begin with finding the number of maximum words by applying the concept of permutation to the above formed table.

$

N = \dfrac{{11!}}{{4! \times 4! \times 2! \times 1!}} \\

\Rightarrow N = \dfrac{{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4!}}{{4! \times 4! \times 2!}} \\

\Rightarrow N = \dfrac{{11 \times 10 \times 9 \times 7 \times 5 \times 48}}{{48}} \\

\Rightarrow N = 34650 \\

$

Let us not look at the question and see what it demands. The demand of the question is to find the number of those words in which 4S’s will be always together. By finding this value, we can easily find the value of k.

Thus, we begin by merging the 4S together and will be considering it as a single letter.

The table shows that the word contains 8 letters in total out of which there is 1 M, 4 I’s, 2P’s and 4S as 1 letter is considered as a single element.

Again, applying the concept of the permutation to the newly formed table we get the total number of required words.

$

\Rightarrow {\text{Total}} = \dfrac{{8!}}{{1! \times 1! \times 4! \times 2!}} \\

= \dfrac{{8 \times 7 \times 6 \times 5 \times 4!}}{{4! \times 2!}} \\

= 8 \times 7 \times 3 \times 5 \\

= 840 \\

$

Now, we know that the probability of any event E, happening is given by;

$P\left( E \right) = \dfrac{{{\text{Favourable outcomes}}}}{{{\text{Total number of outcomes}}}}$

Thus, this gives;

$

P\left( {4S{\text{ coming consecutively }}} \right) = \dfrac{{840}}{{34650}} \\

= \dfrac{4}{{165}} \\

$

It is now given in the question that the probability for four s’s come consecutively in the word ‘MISSISSIPPI’ is $\dfrac{k}{{165}}$.

Thus, by comparing the probability we got and the probability that given in the question, we get that the value of k is 4.

Note: As the question states to find the value of k and k is in the value of the probability of four S’s coming consecutively in the word ‘MISSISSIPPI’. Thus, from here you should be able to understand that you need to first find the probability first and then compare. Don’t try to do the other way around and in that case you might get multiple answers or the wrong one. Also, a crucial step is to make sure to fix 4S’s together and make it a single identity. In the formula for Probability you need to make sure you need to find the total outcomes, which means the total number of words which can be formed using all the 11 letters in the given words without using the condition of 4S’s coming consecutively.

__Complete step-by-step solution__Let us start by considering the given word ‘MISSISSIPPI’. Let us count the number of each letter occurring in the given word.

Total Number of letters | 11 |

Number of ‘S’ | 4 |

Number of ‘I’ | 4 |

Number of ‘P’ | 2 |

Number of ‘M’ | 1 |

Now, we will use the concept of permutation to find out the maximum number of words that can be made with the letters in ‘MISSISSIPPI’. This number would be in turn our maximum number of possible outcomes.

Now, let us begin with finding the number of maximum words by applying the concept of permutation to the above formed table.

$

N = \dfrac{{11!}}{{4! \times 4! \times 2! \times 1!}} \\

\Rightarrow N = \dfrac{{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4!}}{{4! \times 4! \times 2!}} \\

\Rightarrow N = \dfrac{{11 \times 10 \times 9 \times 7 \times 5 \times 48}}{{48}} \\

\Rightarrow N = 34650 \\

$

Let us not look at the question and see what it demands. The demand of the question is to find the number of those words in which 4S’s will be always together. By finding this value, we can easily find the value of k.

Thus, we begin by merging the 4S together and will be considering it as a single letter.

4S | M | I | I | I | I | P | P |

The table shows that the word contains 8 letters in total out of which there is 1 M, 4 I’s, 2P’s and 4S as 1 letter is considered as a single element.

Again, applying the concept of the permutation to the newly formed table we get the total number of required words.

$

\Rightarrow {\text{Total}} = \dfrac{{8!}}{{1! \times 1! \times 4! \times 2!}} \\

= \dfrac{{8 \times 7 \times 6 \times 5 \times 4!}}{{4! \times 2!}} \\

= 8 \times 7 \times 3 \times 5 \\

= 840 \\

$

Now, we know that the probability of any event E, happening is given by;

$P\left( E \right) = \dfrac{{{\text{Favourable outcomes}}}}{{{\text{Total number of outcomes}}}}$

Thus, this gives;

$

P\left( {4S{\text{ coming consecutively }}} \right) = \dfrac{{840}}{{34650}} \\

= \dfrac{4}{{165}} \\

$

It is now given in the question that the probability for four s’s come consecutively in the word ‘MISSISSIPPI’ is $\dfrac{k}{{165}}$.

Thus, by comparing the probability we got and the probability that given in the question, we get that the value of k is 4.

Note: As the question states to find the value of k and k is in the value of the probability of four S’s coming consecutively in the word ‘MISSISSIPPI’. Thus, from here you should be able to understand that you need to first find the probability first and then compare. Don’t try to do the other way around and in that case you might get multiple answers or the wrong one. Also, a crucial step is to make sure to fix 4S’s together and make it a single identity. In the formula for Probability you need to make sure you need to find the total outcomes, which means the total number of words which can be formed using all the 11 letters in the given words without using the condition of 4S’s coming consecutively.

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