Question

In how many ways can the letters in ‘MISSISSIPPI’ be arranged?

Answer 1

Hint: We have permutations and combinations here. it means we are going to rearrange letters by taking one letter or all letters. But we have to be careful as there are many repeated letters. So there will be repeated words as well. Generally when there n different letters in a word, the total number of ways in which all the letters are rearranged is n! ways.

Complete step-by-step solution:
In the word MISSISSIPPI, there are 4 I’s, 2 P’s, 4 S’s.
And the total number of letters including the repetitions is 11 letters.
So the total number of ways in which it can arrange is 11!. But we have to account for all the repeated letters.
So now we have to divide it by 4!, 2!, 4!.
We have to pretend that there are no repeated letters. And then just basically divide it by 4!, 2!, 4! To eliminate all the words with repeated letters
There is a formula for this that is available in permutations and combinations. It states the following :
The number of permutation of n things taken all at a time, in which p things are alike of one kind q is alike of the second kind and r things are alike of the third kind and rest are different is :
$\Rightarrow {\displaystyle \frac{n!}{p!q!r!}}$
So we get the number of ways to arrange MISSISSIPPI is ${\displaystyle \frac{11!}{4!2!4!}}$.
And on further simplification, we get
$\Rightarrow {\displaystyle \frac{11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{(4\times 3\times 2\times 1)(2\times 1)(4\times 3\times 2\times 1)}}$
Now we know that the same terms from numerator and denominator cancels out. Therefore, we get
$\Rightarrow 11\times 10\times 9\times 7\times 5$
$\Rightarrow 34650$
$\therefore$ Hence the number of ways can the letters in ‘MISSISSIPPI’ be arranged is 34650.

Note: Permutations and Combinations is a very tricky chapter and one needs a lot of practice for it. There is a lot of logic and understanding required to solve this kind of question in the limited amount of time. Though there are formulae, it is important to understand the logic behind the question rather than memorizing the formulae. In this way, though we may have trouble solving the first few questions , the rest would gradually become simple.