Introduction
To determine how many 4-letter words can be formed from the letters of the word "PROBLEM," we first analyze the letters available. The word contains 7 distinct letters: P, R, O, B, L, E, M.
Calculating Total 4-Letter Words
- We can choose any 4 letters from the 7 distinct letters.
- The total combinations of 4 letters can be calculated using permutation since the order matters.
Formula for Permutation
- The formula for permutations of n distinct items taken r at a time is: nPr = n! / (n - r)!
- Here, n = 7 (total letters) and r = 4 (letters to choose).
Calculating Permutations
- 7P4 = 7! / (7 - 4)! = 7! / 3! = (7 × 6 × 5 × 4) = 840
Thus, 840 different 4-letter words can be formed from "PROBLEM."
Words Starting and Ending with a Vowel
- The vowels in "PROBLEM" are O and E.
- A 4-letter word must start with a vowel and end with a vowel.
Vowel Positions
1. First Letter (Vowel): Choose either O or E (2 options).
2. Last Letter (Vowel): Choose the remaining vowel (1 option).
3. Middle Letters: Choose 2 letters from the remaining 5 consonants (P, R, B, L, M).
Calculating Middle Letters
- Selecting 2 from 5 can be done in: 5C2 = 10 ways.
- Each selection can be arranged in 2! = 2 ways.
Final Calculation
- Total combinations = 2 (vowels) × 1 (remaining vowel) × 10 (combinations) × 2 (arrangements) = 40.
Conclusion
Thus, there are 40 4-letter words that can be formed from "PROBLEM" that start and end with a vowel.