How many five-letter words beginning and ending with a vowel, without any letter repeated, can be formed from the word “personnel”?
The correct answer and explanation is:
To solve this problem, we will first identify the available letters in the word “personnel,” then calculate how many five-letter words can be formed that begin and end with a vowel, with no repeated letters.
Step 1: Identifying Available Letters
The word “personnel” contains the following distinct letters: p, e, r, s, o, n, l. The vowels in the word are e, o, and e (with e repeating). Therefore, the unique vowels are e and o.
Step 2: Conditions for the Word
- The word must begin and end with a vowel.
- No letter can be repeated in the word.
Step 3: Choosing the First and Last Letters
We have two distinct vowels (e and o) available for the first and last positions. The possible choices for the first and last letters are:
- Choose e for the first position, and o for the last position.
- Choose o for the first position, and e for the last position.
Thus, there are 2 possible ways to choose the first and last vowels.
Step 4: Choosing the Remaining Three Letters
Once the first and last letters are selected, we must pick 3 letters from the remaining available consonants and vowels. The remaining letters from the word “personnel” are: p, r, s, n, l. There are 5 distinct consonants, and these are the only letters available for the middle three positions.
For the middle three positions, we need to select and arrange 3 letters from the 5 remaining consonants. The number of ways to choose and arrange 3 letters out of 5 is given by the permutation formula: P(5,3)=5!(5−3)!=5×4×31=60P(5, 3) = \frac{5!}{(5 – 3)!} = \frac{5 \times 4 \times 3}{1} = 60
Step 5: Total Number of Words
Now, we multiply the number of ways to choose the vowels for the first and last positions (2 ways) by the number of ways to arrange the remaining 3 letters (60 ways): Total number of words=2×60=120\text{Total number of words} = 2 \times 60 = 120
Final Answer:
Thus, 120 distinct five-letter words can be formed from the word “personnel” that begin and end with a vowel, without any repeated letters.