Hint:As we know that the above question is of permutations. We can see that in the word MISSISSIPPI there are four I’s, four S’s, two P’s and one M. We have to find the number of ways the word can be arranged. Using the concept of permutations we can identify the formula and get the number of ways. The basic formula that can be applied in permutations is $\dfrac{{n!}}{{{p_1}!{p_2}!{p_3}!...}}$.
Complete step by step solution:
We can see that in the word MISSISSIPPI there are total $11$ alphabets i.e. $n = 11$ which they can be arranged in total $11$ ways. It means the number of possible ways of $11$ alphabets is $11!$.
But as we can see that there are $4I,4S,2P$, they all can be arranged in $4!,4!,2!$, since these are repeated letters. So ${p_1} = 4,{p_2} = 4,{p_3} = 2$. By substituting the values in the formula the number of arrangements possible with the given alphabets are $\dfrac{{11!}}{{4! \times 4! \times
2!}}$, simplifying this we get $\dfrac{{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5
\times 4!}}{{4! \times 4 \times 3 \times 2 \times 1 \times 2 \times 1}} = \dfrac{{11 \times 10
\times 9 \times 8 \times 7 \times 6 \times 5}}{{4 \times 3 \times 2 \times 1 \times 2 \times 1}}$
It gives the number of ways which is $34650$.
Hence the total number of possible permutations in the word MISSISSIPPI are $34650$.
Note: We should know that permutations means arrangement which is simply the grouping of objects according to the requirement. We should be careful while calculating the number of ways in the word since there are few letters that are repeated and they need to be calculated together. Whenever we face such types of problems the value point to remember is that we need to have a good knowledge of permutations and its formulas.
I'm an enthusiast in combinatorics and permutations, with a demonstrable understanding of the topic. I've explored various permutations problems and applied the principles of combinatorics to solve them effectively. Let's delve into the concept used in the provided article, breaking down each step for a comprehensive understanding.
The article discusses the calculation of permutations for the word MISSISSIPPI, considering the repetition of certain letters. The fundamental concept involved here is the permutation formula:
[ \dfrac{{n!}}{{p_1! \times p_2! \times p_3! \times \ldots}} ]
Where:
- ( n ) is the total number of items (in this case, alphabets in the word).
- ( p_1, p_2, p_3, \ldots ) are the counts of repetitions for each distinct item.
Let's apply this concept to the word MISSISSIPPI:
-
Identifying Parameters:
- Total alphabets (( n )): 11
- Repeated letters: 4 I's, 4 S's, 2 P's
- So, ( p_1 = 4, p_2 = 4, p_3 = 2 )
-
Applying Permutation Formula: [ \dfrac{{11!}}{{4! \times 4! \times 2!}} ]
-
Simplifying the Formula: [ \dfrac{{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4!}}{{4! \times 4 \times 3 \times 2 \times 1 \times 2 \times 1}} ]
-
Further Simplification: [ \dfrac{{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5}}{{4 \times 3 \times 2 \times 1 \times 2 \times 1}} ]
-
Result: The final calculation yields the number of permutations, which is ( 34650 ).
In summary, the total number of possible permutations for the word MISSISSIPPI, considering the repeated letters, is ( 34650 ). This problem highlights the importance of understanding permutations and the need to account for repeated elements in such calculations. The article emphasizes the significance of being meticulous when dealing with repetitions and underscores the necessity of a solid grasp of permutation concepts and formulas.
