# Questionbank: Permutation and Probability

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Simple Rearrangment Question!!

## MANANA Rearrangment

Q.29: In how many ways can we rearrange the letters of the word MANANA such that no two A’s are adjacent to each other?
1. 3 ! * $^{5}C_{3}$
2. 4 ! * $^{4}C_{3}$
3. 3 ! * $^{4}C_{3}$
4. 5 ! * $^{4}C_{3}$

Choice C. 3 ! * $^{4}C_{3}$

## Detailed Solution

MANANA has 6 letters, of which 3 are A's
Now, let us place the letters that are not As on a straight line. We have MNN. These can be arranged in 3! ways.

Now let us create slots between these letters to place the As in.
In order to ensure that no two As are adjacent to each other, let us create exactly one slot between any two letters.
M __N __ N

Additionally, let us add one slot at the beginning and end as well as the As can go there also.
__ M __N __ N __
Now, out of these 4 slots, some 3 can be A. That can be selected in 4C3 ways.

So, total number of words = 3 ! * $^{4}C_{3}$ ways

Correct Answer : 3 ! * $^{4}C_{3}$

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