Answer the questions based on the following information.
Hirawala is a uber rich guy who lives in a swanky place, called Hiranandini, in Mumbai. He has 15 diamond rings and 5 daughters.
If the rings are identical, find the number of ways of distributing these rings among his 5 daughters such that a daughter may get zero or all the rings.

Options

A. (A) 38
B. (B) 76
C. (C) 3876
D. (D) 34

Answer

To solve the problem of distributing 15 identical diamond rings among 5 daughters, we can use the "stars and bars" theorem from combinatorics. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We need to distribute 15 identical rings to 5 daughters. Each daughter can receive any number of rings, including zero.

### Step 2: Set Up the Equation
Let \( x_1, x_2, x_3, x_4, x_5 \) represent the number of rings received by each of the 5 daughters. We need to solve the equation:
\[ x_1 + x_2 + x_3 + x_4 + x_5 = 15 \]
where \( x_i \geq 0 \) for all \( i \).

### Step 3: Apply the Stars and Bars Theorem
The stars and bars theorem states that the number of non-negative integer solutions to the equation \( x_1 + x_2 + ... + x_r = n \) is given by:
\[ \binom{n + r - 1}{r - 1} \]
In our case, \( n = 15 \) (the number of rings) and \( r = 5 \) (the number of daughters).

### Step 4: Substitute the Values
Substituting \( n \) and \( r \) into the formula:
\[ \text{Number of ways} = \binom{15 + 5 - 1}{5 - 1} = \binom{19}{4} \]

### Step 5: Calculate \( \binom{19}{4} \)
Now, we need to calculate \( \binom{19}{4} \):
\[ \binom{19}{4} = \frac{19!}{4!(19-4)!} = \frac{19!}{4! \cdot 15!} \]
This simplifies to:
\[ \binom{19}{4} = \frac{19 \times 18 \times 17 \times 16}{4 \times 3 \times 2 \times 1} \]

### Step 6: Perform the Calculation
Calculating the numerator:
\[ 19 \times 18 = 342 \]
\[ 342 \times 17 = 5814 \]
\[ 5814 \times 16 = 93024 \]
Now, calculating the denominator:
\[ 4! = 24 \]
Now, divide:
\[ \frac{93024}{24} = 3876 \]

### Step 7: Conclusion
Thus, the number of ways to distribute the 15 identical diamond rings among the 5 daughters is:
\[ \boxed{3876} \]