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The question is from Permutation and Combination. This is about Rearrangement of Letters. Our task is to find the probability of a word satisfying particular condition. This section hosts a number of questions which are on par with CAT questions in difficulty on CAT Permutation and Combination, and CAT Probability.

Question 18: If all the rearrangements of the word AMAZON are considered, what is the probability that M will feature between the 2As?

  1. \\frac{1}{3}\\)
  2. \\frac{1}{6}\\)
  3. \\frac{2}{5}\\)
  4. \\frac{3}{8}\\)

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Explanatory Answer

Method of solving this CAT Question from Permutation and Combination: If you know the different rearrangements of letters in the given word, you are through!!

For this type of question, we need to consider only the internal arrangement within the M and 2As.
M and 2As can be rearranged as AMA, AAM, or MAA.
So, the probability that M will feature between the 2As is \\frac{1}{3}\\).

Now, let us think why we need to consider only the M and 2As.
Let us start by considering a set of words where the M and 2 As are placed at positions 2, 3 and 5.
The other three letters have to be in slots 1, 4 and 6
Three letters can be placed in three different slots in 3! = 6 ways.

Now with __ M A __ A __ there are 6 different words.
With __ A M __ A __ there are 6 different words.
With __ A A __ M __ there are 6 different words.
For each selection of the positions for A,A and M, exactly one-third of words will have M between the two A’s.
This is why only the internal arrangement between A, A and M matters.
So, probability of M being between 2 As is \\frac{1}{3}\\).

The question is "what is the probability that M will feature between the 2As?"

Hence the answer is "\\frac{1}{3}\\)"

Choice A is the correct answer.


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