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If the number of ways in which n different things can be distributed among n persons so that at least one person does not get any thing is 232 then what is the value of vfl
- a)3
- b)4
- c)5
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If the number of ways in which n different things can be distributed a...
The number of ways in which n things are distributed among n persons such that every person gets one thing is n!
The total number of ways of distributing n things among n people where each person can get any number of things is nn
Therefore, the total number of ways in which n things are distributed among n persons such that at least one person gets nothing is nn−n!
∴nn−n!=232
There is no general method of solving this equation. By trial and error
22−2!=2
33−3!=21
44−4!=232
∴n=4
Most Upvoted Answer
If the number of ways in which n different things can be distributed a...
Given: The number of ways in which n different things can be distributed among n persons so that at least one person does not get any thing is 232.
To find: The value of vfl.
Solution:
Let's assume that n different things are distributed among n persons in such a way that each person gets at least one thing. In this case, we can use the formula for derangements to calculate the number of ways of distribution.
Derangement: A derangement of a set of n elements is a permutation of the set such that no element appears in its original position.
Formula for derangement: !n = n!(1/0! - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)
Using the above formula, the number of ways of distributing n different things among n persons in such a way that each person gets at least one thing is:
!n = n!(1/0! - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)
Now, the given problem states that at least one person does not get any thing. Let's assume that k persons do not get any thing. We can distribute n different things among (n-k) persons in (n-k)! ways. Now, we need to distribute the remaining k things among the k persons who did not get anything. This can be done in k! ways.
Therefore, the total number of ways of distributing n different things among n persons such that at least one person does not get any thing is:
n! ∑ (-1)^k/k! * (n-k)! * k!
We are given that this value is equal to 232. Let's substitute the given value in the above formula and solve for n.
n! ∑ (-1)^k/k! * (n-k)! * k! = 232
On solving for n, we get n = 4.
Therefore, the value of vfl is 4, which is option (b).
To find: The value of vfl.
Solution:
Let's assume that n different things are distributed among n persons in such a way that each person gets at least one thing. In this case, we can use the formula for derangements to calculate the number of ways of distribution.
Derangement: A derangement of a set of n elements is a permutation of the set such that no element appears in its original position.
Formula for derangement: !n = n!(1/0! - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)
Using the above formula, the number of ways of distributing n different things among n persons in such a way that each person gets at least one thing is:
!n = n!(1/0! - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)
Now, the given problem states that at least one person does not get any thing. Let's assume that k persons do not get any thing. We can distribute n different things among (n-k) persons in (n-k)! ways. Now, we need to distribute the remaining k things among the k persons who did not get anything. This can be done in k! ways.
Therefore, the total number of ways of distributing n different things among n persons such that at least one person does not get any thing is:
n! ∑ (-1)^k/k! * (n-k)! * k!
We are given that this value is equal to 232. Let's substitute the given value in the above formula and solve for n.
n! ∑ (-1)^k/k! * (n-k)! * k! = 232
On solving for n, we get n = 4.
Therefore, the value of vfl is 4, which is option (b).
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