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24 Mar 2023, 14:15
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
61% (01:51) correct
39%
(02:02)
wrong
based on 146
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In how many ways can the letters of the word "GRAPHITE" be rearranged such that the order in which the vowels appear does not change?
A. 8P3
B. 8C3 * 3!
C. 8C3
D. 8C3 * 5!
E. 8C3 * 3! * 5!
A. 8P3
B. 8C3 * 3!
C. 8C3
D. 8C3 * 5!
E. 8C3 * 3! * 5!
24 Mar 2023, 23:23
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Suvankar8250
There are 3 vowels and their relative positions should stay the same (A ... I ... E) . This means there is only 1 way of arranging the 3 vowels within themselves, not 3! ways
Consider an arrangement of consonants and vowels as: GHRPAIET
There would be 3! such arrangements of consonants: GHRPAIET, GHRPAEIT, GHRPIEAT, GHRPIAET, GHRPEAIT, GHRPEIAT
But only 1 of them is acceptable GHRPAIET.
The same will be true for all arrangements. Hence total arrangements of 8 letters is 8! but it should be divided by 3! because out of every 3! cases, only 1 is acceptable. Hence answer is \(\frac{8!}{3!}\) which is the same as 8C3 * 5!
Answer (D)
Alternatively, you can think of it as replacing all vowels by V.
GRVPHVTV
Now arrange: RGVPHVTV, PHGRVVTV ... etc
The 3 Vs are just A... I...E in that order in each case and hence represent 1 arrangement only.
Hence answer will be \(\frac{8!}{3!}\) since it represents the number of ways of arranging GRVPHVTV.
General Discussion
24 Mar 2023, 22:30
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Suvankar8250
Question: " ..... be rearranged such that the order in which the vowels appear does not change"
Inference: 'A' must come before 'I' and 'I' must come before 'E'. Thus the relative order of A, I, and E should ... A..I..E..
Note: These alphabets need not come together or should maintain their original position; however, we need to maintain the relative order.
Total possible arrangements of the word GRAPHITE = 8! The total arrangements also include all possible arrangements of the alphabets A, I, and E. We need to de-arrange these.
Hence the possible arrangements of the word "GRAPHITE" such that the order in which the vowels appear does not change \(8! * \frac{1}{3!}\)
= 8 * 7 * 6 * 5 * 4
This is nothing but \(^8C_3 * 5!\)
Option D
