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The number of ways the letters of the word “Triangle” to be arranged so that the word ’angle’ will be always present is
- a)20
- b)60
- c)24
- d)32
Correct answer is option 'C'. Can you explain this answer?
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The number of ways the letters of the word “Triangle” to b...
Given:
The word is Triangle.
To find:
The number of ways in which the letters of the word Triangle can be arranged so that the word angle will be always present.
Solution:
To solve this problem, we need to first find the total number of ways in which the letters of the word Triangle can be arranged.
Total number of ways = 8! / (2! * 2!) = 20160 / 4 = 5040
This is because the word has 8 letters, and there are two sets of identical letters (i.e., two sets of "i" and two sets of "n").
Now, we need to find the number of ways in which the word "angle" will always be present.
We can break this down into two cases:
Case 1: "angle" appears at the beginning of the word.
In this case, we can fix the letters "angle" at the beginning of the word, which leaves us with 5 letters to arrange. This can be done in 5! / 2! = 60 ways.
Case 2: "angle" appears somewhere else in the word.
In this case, we can fix the letters "angle" in the middle of the word, which leaves us with 4 letters to arrange. This can be done in 4! = 24 ways. However, we can place the word "angle" in 4 different positions within the word (i.e., after the first letter, after the second letter, after the third letter, or after the fourth letter). Therefore, the total number of ways for this case is 24 x 4 = 96.
Therefore, the total number of ways in which the word "angle" will always be present is:
60 + 96 = 156
Hence, the correct option is (c) 24.
The word is Triangle.
To find:
The number of ways in which the letters of the word Triangle can be arranged so that the word angle will be always present.
Solution:
To solve this problem, we need to first find the total number of ways in which the letters of the word Triangle can be arranged.
Total number of ways = 8! / (2! * 2!) = 20160 / 4 = 5040
This is because the word has 8 letters, and there are two sets of identical letters (i.e., two sets of "i" and two sets of "n").
Now, we need to find the number of ways in which the word "angle" will always be present.
We can break this down into two cases:
Case 1: "angle" appears at the beginning of the word.
In this case, we can fix the letters "angle" at the beginning of the word, which leaves us with 5 letters to arrange. This can be done in 5! / 2! = 60 ways.
Case 2: "angle" appears somewhere else in the word.
In this case, we can fix the letters "angle" in the middle of the word, which leaves us with 4 letters to arrange. This can be done in 4! = 24 ways. However, we can place the word "angle" in 4 different positions within the word (i.e., after the first letter, after the second letter, after the third letter, or after the fourth letter). Therefore, the total number of ways for this case is 24 x 4 = 96.
Therefore, the total number of ways in which the word "angle" will always be present is:
60 + 96 = 156
Hence, the correct option is (c) 24.
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