If 25 identical things be distributed among 5 person then
The number of ways each receives atleast one is

Options

A. `.^(24)C_(4)-5`
B. `.^(24)C_(4)-5.^(13)C_(4)`
C. `.^(24)C_(4)-5.^(13)C_(4)`
D. `.^(24)C_(4)`

Answer

To solve the problem of distributing 25 identical things among 5 persons such that each person receives at least one, we can follow these steps:

### Step 1: Understand the problem
We need to distribute 25 identical items to 5 persons, ensuring that each person receives at least one item.

### Step 2: Initial distribution
Since each of the 5 persons must receive at least one item, we start by giving one item to each person. This means we distribute 5 items initially.

### Step 3: Calculate remaining items
After giving one item to each of the 5 persons, we have:
\[ 25 - 5 = 20 \]
So, we have 20 identical items left to distribute.

### Step 4: Set up the equation
Now, we need to find the number of ways to distribute these 20 identical items among the 5 persons. Let \( x_1, x_2, x_3, x_4, x_5 \) be the number of additional items received by each person. We need to solve the equation:
\[ x_1 + x_2 + x_3 + x_4 + x_5 = 20 \]
where \( x_i \geq 0 \) for \( i = 1, 2, 3, 4, 5 \).

### Step 5: Use the stars and bars theorem
To solve this equation, we can use the "stars and bars" theorem, which states that the number of ways to distribute \( n \) identical items into \( r \) distinct groups is given by:
\[ \binom{n + r - 1}{r - 1} \]
In our case, \( n = 20 \) (the remaining items) and \( r = 5 \) (the persons).

### Step 6: Apply the formula
Substituting the values into the formula, we get:
\[ \binom{20 + 5 - 1}{5 - 1} = \binom{24}{4} \]

### Step 7: Final answer
Thus, the number of ways to distribute 25 identical things among 5 persons, ensuring that each person receives at least one item, is:
\[ \binom{24}{4} \]