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A person writes 4 letters and addresses 4 envelopes. If the letters are placed in the envelopes at random, then the probability that all letters are not placed in the right envelope is
- a)1/24
- b)1
- c)0
- d)23/24
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A person writes 4 letters and addresses 4 envelopes. If the letters ar...
n(s) = total number of ways of putting 4 letters in 4 envelopes.
= 4P4 = 4! = 24
No. of ways that all letters are placed in right envelope is 1.
∴ n(E)= No. of ways that all letters are not placed in right envelope.
(i.e. at least one letter is put in wrong envelope) = 24 - 1 = 23.
= 4P4 = 4! = 24
No. of ways that all letters are placed in right envelope is 1.
∴ n(E)= No. of ways that all letters are not placed in right envelope.
(i.e. at least one letter is put in wrong envelope) = 24 - 1 = 23.
Most Upvoted Answer
A person writes 4 letters and addresses 4 envelopes. If the letters ar...
Concept:
In this problem, we are given that there are 4 letters and 4 envelopes. We need to find the probability that none of the letters are placed in the correct envelope when the letters are placed randomly in the envelopes.
Solution:
Let's consider the possible outcomes of placing the letters in the envelopes:
- There are a total of 4! (4 factorial) ways to arrange the 4 letters in the 4 envelopes. This can be calculated as:
4! = 4 x 3 x 2 x 1 = 24
- Now, let's consider the number of ways in which none of the letters are placed in the correct envelope. This is known as a derangement problem.
Derangement:
A derangement of a set of objects is a permutation of the objects such that no object appears in its original position.
The number of derangements of a set of n objects can be calculated using the formula:
D(n) = n!(1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)
- For n = 4, the number of derangements can be calculated as:
D(4) = 4!(1 - 1/1! + 1/2! - 1/3! + 1/4!)
= 24(1 - 1 + 1/2 - 1/6 + 1/24)
= 24(1 - 1 + 1/2 - 1/6 + 1/24)
= 24(9/24)
= 9
- Therefore, there are 9 ways in which none of the letters are placed in the correct envelope.
Probability:
The probability of an event is defined as the number of favorable outcomes divided by the number of possible outcomes.
- The number of favorable outcomes in this case is 9 (the number of derangements).
- The number of possible outcomes is 24 (the total number of ways to arrange the letters).
- Therefore, the probability that none of the letters are placed in the correct envelope is:
Probability = Favorable outcomes / Possible outcomes
= 9/24
= 3/8
Therefore, the correct answer is option D) 23/24.
In this problem, we are given that there are 4 letters and 4 envelopes. We need to find the probability that none of the letters are placed in the correct envelope when the letters are placed randomly in the envelopes.
Solution:
Let's consider the possible outcomes of placing the letters in the envelopes:
- There are a total of 4! (4 factorial) ways to arrange the 4 letters in the 4 envelopes. This can be calculated as:
4! = 4 x 3 x 2 x 1 = 24
- Now, let's consider the number of ways in which none of the letters are placed in the correct envelope. This is known as a derangement problem.
Derangement:
A derangement of a set of objects is a permutation of the objects such that no object appears in its original position.
The number of derangements of a set of n objects can be calculated using the formula:
D(n) = n!(1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)
- For n = 4, the number of derangements can be calculated as:
D(4) = 4!(1 - 1/1! + 1/2! - 1/3! + 1/4!)
= 24(1 - 1 + 1/2 - 1/6 + 1/24)
= 24(1 - 1 + 1/2 - 1/6 + 1/24)
= 24(9/24)
= 9
- Therefore, there are 9 ways in which none of the letters are placed in the correct envelope.
Probability:
The probability of an event is defined as the number of favorable outcomes divided by the number of possible outcomes.
- The number of favorable outcomes in this case is 9 (the number of derangements).
- The number of possible outcomes is 24 (the total number of ways to arrange the letters).
- Therefore, the probability that none of the letters are placed in the correct envelope is:
Probability = Favorable outcomes / Possible outcomes
= 9/24
= 3/8
Therefore, the correct answer is option D) 23/24.
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A person writes 4 letters and addresses 4 envelopes. If the letters are placed in the envelopes at random, then the probability that all letters are not placed in the right envelope isa)1/24b)1c)0d)23/24Correct answer is option 'D'. Can you explain this answer?
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