Question

The English alphabets have 5 vowels and 21 consonants. How many words with two different vowels and two different consonants can be formed from the alphabet?

• A. \(2100 \times 2!\)
• B. \(210 \times 2!\)
• C. \(210 \times 4!\)
• D. \(2100 \times 4!\)

Hint:

When a question involves selecting letters and then arranging them, first use combinations to choose the required letters and then use permutations to arrange them. \[ \text{Total Ways} = \text{Selection Ways} \times \text{Arrangement Ways} \]

Answer 1

The Correct Option is D

Step 1: Choose 2 vowels and 2 consonants.
Number of ways to choose 2 vowels from 5: \(\binom{5}{2}=10\). Number of ways to choose 2 consonants from 21: \(\binom{21}{2}=210\). Total selections: \(10 \times 210 = 2100\).

Step 2: Arrange the 4 letters.
The 4 chosen letters can be arranged in \(4! = 24\) ways. Total words: \(2100 \times 4!\).
\[\boxed{2100 \times 4!}\]