If 6 objects are distributed at random among 6 persons, the probability that atleast one of them will not get any thing is :

Options

A. `6.(6!)`
B. `(5^6)/(6!)`
C. `(6^6 -6!)/(6^6)`
D. none of these

Answer

To find the probability that at least one of the 6 persons will not receive any object when 6 objects are distributed randomly among them, we can use the concept of complementary probability. Here’s the step-by-step solution:

### Step 1: Calculate the Total Ways to Distribute 6 Objects
When distributing 6 objects among 6 persons, each object can go to any of the 6 persons. Therefore, the total number of ways to distribute the 6 objects is given by:

\[ \text{Total Ways} = 6^6 \]

### Step 2: Calculate the Ways Where Everyone Gets At Least One Object
To find the probability that at least one person does not get an object, we first find the probability that everyone gets at least one object. This can be calculated using the principle of inclusion-exclusion or by recognizing it as a permutation problem.

The number of ways to distribute 6 objects such that each person gets at least one object is equivalent to the number of surjective functions from a set of 6 objects to a set of 6 persons. This is given by \(6!\) (6 factorial), which is the number of ways to assign each object to a different person.

\[ \text{Ways where everyone gets at least one object} = 6! \]

### Step 3: Calculate the Probability that Everyone Gets At Least One Object
Now we can find the probability that everyone gets at least one object:

\[ P(\text{Everyone gets at least one}) = \frac{6!}{6^6} \]

### Step 4: Calculate the Probability that At Least One Person Does Not Get an Object
To find the probability that at least one person does not get an object, we can use the complement of the previous probability:

\[ P(\text{At least one does not get an object}) = 1 - P(\text{Everyone gets at least one}) \]

Substituting the values we calculated:

\[ P(\text{At least one does not get an object}) = 1 - \frac{6!}{6^6} \]

### Step 5: Simplify the Expression
Now we can simplify the expression:

\[ P(\text{At least one does not get an object}) = 1 - \frac{720}{6^6} \]

Calculating \(6^6\):

\[ 6^6 = 46656 \]

So,

\[ P(\text{At least one does not get an object}) = 1 - \frac{720}{46656} \]

Calculating the fraction:

\[ \frac{720}{46656} \approx 0.0154321 \]

Thus,

\[ P(\text{At least one does not get an object}) \approx 1 - 0.0154321 \approx 0.9845679 \]

### Final Answer
The probability that at least one of the 6 persons will not get anything is approximately:

\[ P(\text{At least one does not get an object}) \approx 0.9846 \]