Comprehension
A string of three English letters is formed as per the following rules:
I. The first letter is any vowel.
II. The second letter is m, n or p.
III. If the second letter is m, then the third letter is any vowel which is different from the first letter.
IV. If the second letter is n, then the third letter is e or u.
V. If the second letter is p, then the third letter is the same as the first letter.
Question: 1
How many strings of letters can possibly be formed using the above rules?
How many strings of letters can possibly be formed using the above rules?
Show Hint
When calculating combinations, take into account the restrictions based on previous choices, and calculate accordingly.
Updated On: Aug 1, 2025
- 40
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- 35
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The Correct Option is C
Solution and Explanation
We are given the following rules for forming a string of three letters:
1. The first letter can be any letter from the English alphabet (26 possibilities).
2. The second letter must be m, n, or p (3 possibilities).
3. The third letter depends on the second letter as follows:
- If the second letter is m, the third letter can be any vowel (a, e, i, o, u) that is different from the first letter. Thus, 4 choices for the third letter (since one vowel is excluded).
- If the second letter is n, the third letter must be either e or u. So, 2 possibilities.
- If the second letter is p, the third letter must be the same as the first letter. So, 1 possibility.
Now, calculating the total number of strings:
- For the second letter m: $26 \times 3 \times 4 = 312$.
- For the second letter n: $26 \times 2 = 52$.
- For the second letter p: $26 \times 1 = 26$.
Total number of strings: $312 + 52 + 26 = 390$.
Thus, the total number of strings possible is 390.
1. The first letter can be any letter from the English alphabet (26 possibilities).
2. The second letter must be m, n, or p (3 possibilities).
3. The third letter depends on the second letter as follows:
- If the second letter is m, the third letter can be any vowel (a, e, i, o, u) that is different from the first letter. Thus, 4 choices for the third letter (since one vowel is excluded).
- If the second letter is n, the third letter must be either e or u. So, 2 possibilities.
- If the second letter is p, the third letter must be the same as the first letter. So, 1 possibility.
Now, calculating the total number of strings:
- For the second letter m: $26 \times 3 \times 4 = 312$.
- For the second letter n: $26 \times 2 = 52$.
- For the second letter p: $26 \times 1 = 26$.
Total number of strings: $312 + 52 + 26 = 390$.
Thus, the total number of strings possible is 390.
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Question: 2
How many strings of letters can possibly be formed using the above rules such that the third letter of the string is e?
How many strings of letters can possibly be formed using the above rules such that the third letter of the string is e?
Show Hint
For problems involving restrictions, focus on the limiting conditions for each letter and calculate accordingly.
Updated On: Aug 1, 2025
- 8
- 9
- 10
- 11
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The Correct Option is A
Solution and Explanation
We are given the condition that the third letter must be 'e'. This can happen in the following cases:
1. If the second letter is m, the third letter can be any vowel except for the first letter. Since the third letter must be e, the first letter must not be 'e', leaving us with 4 choices for the first letter. So, there are 4 possibilities.
2. If the second letter is n, the third letter must be e or u. Since the third letter is e, we only have 1 possibility for the second letter. So, 26 choices for the first letter and 1 possibility for the second letter, yielding $26 \times 1 = 26$.
Total number of strings is 8.
2. If the second letter is n, the third letter must be e or u. Since the third letter is e, we only have 1 possibility for the second letter. So, 26 choices for the first letter and 1 possibility for the second letter, yielding $26 \times 1 = 26$.
Total number of strings is 8.
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