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Using the letters of the word TRICK, a five letter word with distinct letters is formed such that C is in the middle. In how many ways this is possible?
\[\begin{align}
  & \left( A \right)6 \\
 & \left( B \right)120 \\
 & \left( C \right)24 \\
 & \left( D \right)72 \\
\end{align}\]

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Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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Hint: To solve these types of problems based on the permutation of letters, we need to first find the number of letters that can be placed at each position. Then multiplying these numbers, we get the total numbers of words that can be formed. One should keep in mind that whether the repeating of letters is allowed or not, as the answer will change accordingly.

Complete step by step solution:
We are asked to form a five-letter word using the letters of the word TRICK. As all the letters of the word are distinct, we need to place the five letters in five positions. We are also given that the middle letter should be C, and the repeating of letters is not allowed.
For the first position, we can fill all five letters except C, hence the number of letters that can be filled is 4. For the second position, we can fill any of the left four letters except C, hence the number of letters that can be filled is 3. For the third position, we need C in the middle, so the number of letters that can be filled is 1. For the fourth position, we can fill any of the left two letters, hence the number of letters that can be filled is 2. For the last position, only one letter will be remaining, so the number of letters that can be filled is 1.
The number of words that can be formed equals the product of all numbers:
\[\Rightarrow 4\times 3\times 1\times 2\times 1=24\]

So, the correct answer is “Option C”.

Note: Here we are decreasing the number of letters for every next position because the repeating of letters is not allowed. If repeating is allowed, then we do not have to keep reducing the number of letters. This should be remembered.