Answer the following questions independently of each other
In how many ways can 12 identical bouquets be kept in 3 distinct crates ?

Options

A. (A) 19
B. (B) 91
C. (C) 81
D. (D) 61

Answer

To solve the problem of distributing 12 identical bouquets into 3 distinct crates, we can use the "stars and bars" theorem from combinatorics. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We need to distribute 12 identical bouquets (let's call them "stars") into 3 distinct crates (let's call them "bars"). We want to find the number of ways to do this.

### Step 2: Set Up the Equation
Let \( X_1 \), \( X_2 \), and \( X_3 \) represent the number of bouquets in crates C1, C2, and C3 respectively. The equation we need to satisfy is:
\[ X_1 + X_2 + X_3 = 12 \]
where \( X_1, X_2, X_3 \geq 0 \).

### Step 3: Apply the Stars and Bars Theorem
According to the stars and bars theorem, the number of non-negative integral solutions to the equation \( X_1 + X_2 + X_3 = N \) is given by:
\[ \binom{N + R - 1}{R - 1} \]
where \( N \) is the total number of items to distribute (in this case, 12 bouquets), and \( R \) is the number of groups (in this case, 3 crates).

### Step 4: Substitute Values
Here, \( N = 12 \) and \( R = 3 \). Plugging these values into the formula gives us:
\[ \binom{12 + 3 - 1}{3 - 1} = \binom{14}{2} \]

### Step 5: Calculate the Binomial Coefficient
Now, we calculate \( \binom{14}{2} \):
\[ \binom{14}{2} = \frac{14 \times 13}{2 \times 1} = \frac{182}{2} = 91 \]

### Step 6: Conclusion
Thus, the number of ways to distribute 12 identical bouquets into 3 distinct crates is **91**.

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