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Find the probability that in a random arrangement of the letters of the word DAUGHTER, the letter D occupies the first place.
- a)1/8
- b)1/4
- c)3/8
- d)1/2
Correct answer is option 'A'. Can you explain this answer?
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Find the probability that in a random arrangement of the letters of t...
If D occupies the first place n(A) = 7!, Total words = n(S) = 8!
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∴ Required probability
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Find the probability that in a random arrangement of the letters of t...
Probability of D occupying the first place in a random arrangement of the letters of the word DAUGHTER
To find the probability, we need to determine the total number of possible arrangements and the number of arrangements in which the letter D occupies the first place.
Total number of arrangements:
The word DAUGHTER has 8 letters. Therefore, there are 8 possible positions for the first letter, 7 possible positions for the second letter, and so on. Hence, the total number of possible arrangements is 8!.
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
Number of arrangements with D in the first place:
Since we want the letter D to occupy the first place, we can fix D in the first position and arrange the remaining 7 letters in the remaining 7 positions. The remaining 7 letters can be arranged in 7! ways.
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040
Probability:
The probability is given by the number of favorable outcomes (arrangements with D in the first place) divided by the total number of possible outcomes.
Probability = Number of arrangements with D in the first place / Total number of possible arrangements
Probability = 5,040 / 40,320
Simplifying the above expression:
Probability = 1/8
Therefore, the probability that in a random arrangement of the letters of the word DAUGHTER, the letter D occupies the first place is 1/8.
Hence, the correct answer is option 'A' (1/8).
To find the probability, we need to determine the total number of possible arrangements and the number of arrangements in which the letter D occupies the first place.
Total number of arrangements:
The word DAUGHTER has 8 letters. Therefore, there are 8 possible positions for the first letter, 7 possible positions for the second letter, and so on. Hence, the total number of possible arrangements is 8!.
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
Number of arrangements with D in the first place:
Since we want the letter D to occupy the first place, we can fix D in the first position and arrange the remaining 7 letters in the remaining 7 positions. The remaining 7 letters can be arranged in 7! ways.
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040
Probability:
The probability is given by the number of favorable outcomes (arrangements with D in the first place) divided by the total number of possible outcomes.
Probability = Number of arrangements with D in the first place / Total number of possible arrangements
Probability = 5,040 / 40,320
Simplifying the above expression:
Probability = 1/8
Therefore, the probability that in a random arrangement of the letters of the word DAUGHTER, the letter D occupies the first place is 1/8.
Hence, the correct answer is option 'A' (1/8).
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Find the probability that in a random arrangement of the letters of the word DAUGHTER, the letter D occupies the first place.a)1/8b)1/4c)3/8d)1/2Correct answer is option 'A'. Can you explain this answer?
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