How Many 5-Letter Words can be Formed Using the Letters of the English Alphabet that Contain 2 Different Vowels and 3 Different Consonants?

Question: How many 5-letter words can be formed using the letters of the English alphabet that contain 2 different vowels and 3 different consonants?

1. 5C2 * 21C3
2. 5P2 * 21P3
3. 5C2 * 21C3 * 5!
4. 5P2 * 21P3 * 5!
5. 5^2 * 21^3 * 5!

“How many 5-letter words can be formed using the letters of the English alphabet that contain 2 different vowels and 3 different consonants?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

The problem statement informs that:
Given:
The letters of the English alphabet that contain

• 2 different vowels
• 3 different consonants

Find Out:

• How many 5-letter words can be formed

Let the arrangement be V1 V2 C1 C2 C3

The V1 and V2 are the vowels (5 in total in English alphabet)
C1,C2,C3 are the consonants (21 in total in English alphabet)

Now we will see how we can select

• 2 vowels out of 5 available in 5C2 ways
• 3 consonants out of 21 in 21C3 ways.

Additionally, the arrangement mentioned above can be arranged in a further 5! ways.

Thus the total number of arrangements = 5!*5C2*21C3.
Hence, C is the correct answer.

Approach Solution 2:

The English alphabets have 5 Vowels A,E,I,O,U and 21 Consonants.

The problem statement says that 2 different vowels and 3 different consonants.
Hence, we understand that repetitions are not allowed.

Note: Additional. if the word "different" is not given,

If it was, the solution would have bee:
Total 5 letter words 5*5*21*21*21

Comming back to the problem provided:

We can choose 2 different vowels in 5C2 ways
We can also choose 3 different consonants in 21C3
Also, we can arrange them in 5! Ways.
Total 5 letter words 5C2∗21C3∗5!

Hence, C is the correct option.

Approach Solution 3:

The problem statement informs that:
Given:
The letters of the English alphabet that contain

• 2 different vowels
• 3 different consonants

Find Out:

• How many 5-letter words can be formed

2 different vowels can be selected from 5 in 5c2 ways
3 different consonants can be selected from 21 in 21c3 ways
Also, we will need to arrange the 5 alphabets in 5 ! ways
Hence, the total number of ways are =5c2*21c3*5!

Hence, C is the correct answer.

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