Answer the questions based on the following information. There are four numbers a,b,c,d such that a+b+c+d=12
Find the number of non-negative integral solutions of the equation.

Options

A. 25
B. 244
C. 455
D. 425

Answer

To find the number of non-negative integral solutions to the equation \( a + b + c + d = 12 \), 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 find the number of ways to distribute 12 identical objects (the total sum) into 4 distinct boxes (the variables \( a, b, c, d \)) where each box can hold zero or more objects.

### Step 2: Apply the Stars and Bars Theorem
The stars and bars theorem states that the number of ways to distribute \( n \) identical objects into \( r \) distinct boxes is given by the formula:
\[ \binom{n + r - 1}{r - 1} \]
In our case, \( n = 12 \) (the total sum) and \( r = 4 \) (the number of variables).

### Step 3: Substitute the Values into the Formula
Using the formula:
\[ \binom{12 + 4 - 1}{4 - 1} = \binom{15}{3} \]

### Step 4: Calculate \( \binom{15}{3} \)
Now, we calculate \( \binom{15}{3} \):
\[ \binom{15}{3} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} \]

### Step 5: Simplify the Calculation
Calculating the numerator:
\[ 15 \times 14 = 210 \]
\[ 210 \times 13 = 2730 \]
Now, calculating the denominator:
\[ 3 \times 2 \times 1 = 6 \]
Now, divide the numerator by the denominator:
\[ \frac{2730}{6} = 455 \]

### Conclusion
Thus, the number of non-negative integral solutions to the equation \( a + b + c + d = 12 \) is \( 455 \).