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# How many four letter words are possible using the first 5 letters of the alphabet if the first letter cannot be $a$ and adjacent letters cannot be alike?

Last updated date: 05th Aug 2024
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Hint: Here, we will find a number of possible ways to fill the first place of the word then while remembering that it cannot be $a$. Then we will follow the same procedure to fill the second place and the rest of the place. Finally, we will multiply all the possible ways to get the required answer.

The first 5 letters of the alphabet are a, b, c, d and e.
But, it is given that the first letter of the word cannot be $a$, so,
The possible way to fill first place of the word $= 4$ ways
Next, the second letter cannot be the same as the first but we can use all the other four letters.
So, the possible way to fill second place of the word $= 4$ ways
Next, the third will not be the same as the first and second.
So, the possible way to fill second place of the word $= 3$ ways
Similarly, the possible way to fill second place of the word $= 2$ ways

So, the total number of ways $= 4 \times 4 \times 3 \times 2 = 96$ ways
A total of 96 four letter words can be formed by the first 5 letters.

Note:
Here, it is mentioned that no two letters should be the same which means repetition is not allowed. So, we can make mistakes if we repeat the digits. Also, it is mentioned that only the first letter cannot be ‘a’, on the other three places we can place ‘a’. The method of arranging elements from a set of elements such that the order of arrangement matters is called a permutation.