Question

How many words each of two vowels and three consonants can be formed from the letters of the word "UNIVERSAL"?

• A. 7000
• B. 7200
• C. 7400
• D. 7800

Hint:

Choose 2 vowels and 3 consonants separately, then arrange all 5 letters.

Answer 1

The Correct Option is B

To determine how many words, each consisting of two vowels and three consonants, can be formed from the letters of the word "UNIVERSAL," we will follow these steps: 

1. Identify the vowels and consonants in the word "UNIVERSAL":
   • Vowels: U, I, E, A (4 vowels)
   • Consonants: N, V, R, S, L (5 consonants)
2. Calculate the number of ways to choose two vowels out of four:
   • Using the combination formula: \(^nC_r = \frac{n!}{r!(n-r)!}\)
   • Number of ways = \(^4C_2 = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6\)
3. Calculate the number of ways to choose three consonants out of five:
   • Number of ways = \(^5C_3 = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10\)
4. Calculate the total number of letter combinations (2 vowels + 3 consonants):
   • Number of combinations = Number of ways to choose vowels × Number of ways to choose consonants
   • Total combinations = 6 × 10 = 60
5. Consider the arrangement of these 5 letters in a word:
   • Each combination of 5 letters can be arranged in 5! ways
   • 5! = 5 × 4 × 3 × 2 × 1 = 120
6. Calculate the final total number of words:
   • Total number of words = Total combinations × Arrangements per combination
   • Total words = 60 × 120 = 7200

Therefore, the correct answer is \(7200\).

Answer 2

Step 1: Split the letters of UNIVERSAL into vowels and consonants.
UNIVERSAL has 9 distinct letters: U, N, I, V, E, R, S, A, L. The vowels are U, I, E, A (4 of them), and the consonants are N, V, R, S, L (5 of them).

Step 2: Choose 2 vowels out of the 4 available.
This is a selection, so we use combinations: \( {}^4C_2 = \dfrac{4!}{2! \times 2!} = 6 \) ways.

Step 3: Choose 3 consonants out of the 5 available.
\( {}^5C_3 = \dfrac{5!}{3! \times 2!} = 10 \) ways.

Step 4: Arrange the 5 chosen letters into a word.
Once we have picked 2 vowels and 3 consonants, that is 5 distinct letters, and they can be arranged in \(5! = 120\) ways to form different words.

Final Answer:
Total words \(= 6 \times 10 \times 120 = 7200\).
\[ \boxed{7200} \]