10 mangoes are to be distributed among 5 persons. The probability that at least one of them will receive none, is

Options

A. `(35)/(143)`
B. `(108)/(143)`
C. `(18)/(143)`
D. `(125)/(143)`

Explanation

To solve the problem of distributing 10 mangoes among 5 persons and finding the probability that at least one of them will receive none, we can follow these steps:

### Step 1: Calculate the total number of ways to distribute 10 mangoes among 5 persons.

We use the formula for distributing \( N \) identical items to \( R \) persons, which is given by:

\[ \binom{N + R - 1}{R - 1} \]

In our case, \( N = 10 \) (mangoes) and \( R = 5 \) (persons). Thus, we calculate:

\[ \text{Total ways} = \binom{10 + 5 - 1}{5 - 1} = \binom{14}{4} \]

### Step 2: Calculate the number of ways to distribute 10 mangoes such that each person receives at least one mango.

If each person must receive at least one mango, we can first give one mango to each of the 5 persons. This uses up 5 mangoes, leaving us with \( 10 - 5 = 5 \) mangoes to distribute freely among the 5 persons.

Now, we need to find the number of ways to distribute these remaining 5 mangoes, which can be calculated as:

\[ \text{Ways with each receiving at least one} = \binom{5 + 5 - 1}{5 - 1} = \binom{9}{4} \]

### Step 3: Calculate the probability that at least one person receives none.

To find the probability that at least one person receives none, we can use the complementary probability approach.

The probability that each person receives at least one mango is given by:

\[ P(\text{each receives at least one}) = \frac{\text{Ways with each receiving at least one}}{\text{Total ways}} = \frac{\binom{9}{4}}{\binom{14}{4}} \]

Thus, the probability that at least one person receives none is:

\[ P(\text{at least one receives none}) = 1 - P(\text{each receives at least one}) = 1 - \frac{\binom{9}{4}}{\binom{14}{4}} \]

### Step 4: Calculate the values of the combinations.

Now we compute the combinations:

1. \(\binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126\)
2. \(\binom{14}{4} = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} = 1001\)

### Step 5: Substitute the values into the probability formula.

Now substituting these values into our probability equation:

\[ P(\text{at least one receives none}) = 1 - \frac{126}{1001} \]

Calculating this gives:

\[ P(\text{at least one receives none}) = \frac{1001 - 126}{1001} = \frac{875}{1001} \]

### Step 6: Simplify the fraction.

The fraction \(\frac{875}{1001}\) can be simplified if possible. However, in this case, it is already in its simplest form.

### Final Answer:

Thus, the probability that at least one of the 5 persons will receive none of the mangoes is:

\[ \frac{875}{1001} \]