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There are 5 letters and 5 directed envelopes. The number of ways in which all the letters be put in a wrong envelope is _______.(Nearest Integer)
Correct answer is '44'. Can you explain this answer?
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There are 5 letters and 5 directed envelopes. The number of ways in w...
Problem:
There are 5 letters and 5 directed envelopes. The number of ways in which all the letters be put in a wrong envelope is _______. (Nearest Integer)
Solution:
To find the number of ways in which all the letters can be put in the wrong envelopes, we need to find the number of derangements of the letters. A derangement is a permutation of the elements of a set, such that no element appears in its original position.
Derangement Formula:
The number of derangements of a set with n elements can be found using the derangement formula:
D(n) = n!(1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)
Applying the Derangement Formula:
In this case, we have 5 letters, so we need to find D(5).
D(5) = 5!(1 - 1/1! + 1/2! - 1/3! + 1/4! - 1/5!)
= 120(1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
= 120(1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
= 120(1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
= 120(1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
= 120(1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
= 44.
Therefore, the number of ways in which all the letters can be put in the wrong envelopes is 44 (nearest integer).
There are 5 letters and 5 directed envelopes. The number of ways in which all the letters be put in a wrong envelope is _______. (Nearest Integer)
Solution:
To find the number of ways in which all the letters can be put in the wrong envelopes, we need to find the number of derangements of the letters. A derangement is a permutation of the elements of a set, such that no element appears in its original position.
Derangement Formula:
The number of derangements of a set with n elements can be found using the derangement formula:
D(n) = n!(1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)
Applying the Derangement Formula:
In this case, we have 5 letters, so we need to find D(5).
D(5) = 5!(1 - 1/1! + 1/2! - 1/3! + 1/4! - 1/5!)
= 120(1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
= 120(1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
= 120(1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
= 120(1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
= 120(1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
= 44.
Therefore, the number of ways in which all the letters can be put in the wrong envelopes is 44 (nearest integer).
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There are 5 letters and 5 directed envelopes. The number of ways in which all the letters be put in a wrong envelope is _______.(Nearest Integer)Correct answer is '44'. Can you explain this answer?
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