How many words can be formed with the letters of the word 'POSTMAN', if every word begins with T and ends with M?
120
Solving the 'POSTMAN' Word Arrangement Problem with Constraints
Let's break down how to solve this word formation problem using the letters of the word 'POSTMAN'. The question asks for the number of distinct words that can be formed from the letters of 'POSTMAN' under specific conditions: every word must begin with the letter 'T' and end with the letter 'M'.
Understanding the Letters in 'POSTMAN'
The word 'POSTMAN' has 7 distinct letters: P, O, S, T, M, A, N. There are no repeated letters in 'POSTMAN'.
Applying the Constraints
The constraints are:
- The first letter must be 'T'.
- The last letter must be 'M'.
Since the word has 7 letters, we have 7 positions to fill.
Position 1 is fixed as 'T'. There is only 1 choice for this position.
Position 7 is fixed as 'M'. There is only 1 choice for this position.
Arranging the Remaining Letters in 'POSTMAN'
After placing 'T' at the beginning and 'M' at the end, we are left with the letters P, O, S, A, N from the original word 'POSTMAN'. There are 5 remaining letters.
These 5 letters need to be arranged in the remaining 5 positions (positions 2, 3, 4, 5, and 6).
The number of ways to arrange 5 distinct letters in 5 distinct positions is given by the number of permutations of 5 items taken 5 at a time, which is 5! (5 factorial).
Calculating the Number of Words
The calculation for 5! is as follows:
\( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)
\( 5! = 20 \times 6 \)
\( 5! = 120 \)
So, there are 120 ways to arrange the remaining 5 letters (P, O, S, A, N) in the middle 5 positions.
Since the first position must be 'T' (1 way) and the last must be 'M' (1 way), the total number of words formed is the product of the number of choices for each position:
Total words = (Choices for Pos 1) × (Arrangement of remaining 5 letters) × (Choices for Pos 7)
Total words = \( 1 \times 5! \times 1 \)
Total words = \( 1 \times 120 \times 1 \)
Total words = \( 120 \)
Therefore, 120 different words can be formed using the letters of 'POSTMAN' such that each word begins with T and ends with M. This type of problem involves calculating permutations under specific constraints.
This method helps solve problems related to arranging letters from words like 'POSTMAN' with fixed starting and ending characters.
Important Questions from Permutations
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