Question: How many three-letter words can be constructed using all the 26 letters of the English alphabet if only the middle letter of each word is a vowel (a, e, i, o, u) and the repetition of the letters is allowed?
- 9261
- 2400
- 2205
- 441
- 105
“How many three-letter words can be constructed using all the 26 letters of the English alphabet”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.
Solution and Explanation:
Approach Solution 1:
The given question requires finding the number of three-letter words that can be formed with 26 letters in the English alphabet. Accordingly, one letter in the middle of the word needs to be a vowel. Further, repetition of the letters is allowed in it.
Accordingly, the first aspect can be understood in terms of having
26 - 5 ways = 21 ways
The second way is the use of the five vowels = 5 ways
The third way of formation of words can be again
26 - 5 ways = 21 ways
Considering that repetition is allowed, it can be evaluated that
21 * 5 * 21 = 2205
Hence, 2205 three-letter words can be constructed from the 26 alphabets and use of vowels which can be repeated.
Correct Answer: C
Approach Solution 2:
The answer to the provided question is to determine how many three-letter words can be constructed using the 26 letters of the English alphabet. As a result, the centre of the word must have a vowel. Additionally, it permits letter repetition.
In light of this, the first aspect might be regarded as having 26 ways – 5 ways = 21 ways.
The second method is to employ all five vowels (five methods in total).
The third method of word construction is possible once more.
26 – 5 = 21 ways.
Given that repetition is permitted, the calculation is 21 * 5 * 21 = 2205.
As a result, using the 26 alphabets and repeatable vowels, 2205 three-letter words may be created.
Correct Answer: C
Approach Solution 3:
How many three-letter words may be made using the 26 letters of the English alphabet is the answer to the given question. As a result, the word's middle vowel is required. It also allows for letter repetition.
This means that the first component might be thought of as having 26 ways, minus 5 ways, or 21 ways.
The second strategy involves using all five vowels (five methods in total).
Once more, the third approach of word formation is viable.
26 – 5 = 21 ways.
The calculation is 21 * 5 * 21 = 2205, assuming that repetition is allowed.
As a consequence, 2205 three-letter words may be made utilising the 26 alphabets and repeating vowels.
Correct Answer: C
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