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 positive integral solutions of the equation.

Options

A. 125
B. 165
C. 143
D. 156

Answer

To find the number of positive integral solutions to the equation \( a + b + c + d = 12 \), we can use the "stars and bars" theorem in 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 items (the total sum) into 4 distinct groups (the variables \( a, b, c, d \)), where each group must contain at least one item since \( a, b, c, d \) are positive integers.

### Step 2: Adjust for positive integers
Since \( a, b, c, d \) must be positive integers, we can redefine the variables to account for the minimum requirement. Let:
- \( a' = a - 1 \)
- \( b' = b - 1 \)
- \( c' = c - 1 \)
- \( d' = d - 1 \)

Now, \( a', b', c', d' \) are non-negative integers (they can be zero). The equation now becomes:
\[ (a' + 1) + (b' + 1) + (c' + 1) + (d' + 1) = 12 \]
This simplifies to:
\[ a' + b' + c' + d' = 8 \]

### Step 3: Apply the stars and bars theorem
We need to find the number of non-negative integer solutions to the equation \( a' + b' + c' + d' = 8 \). According to the stars and bars theorem, the number of solutions is given by the formula:
\[ \binom{n + r - 1}{r - 1} \]
where \( n \) is the total number of items to distribute (8 in this case), and \( r \) is the number of groups (4 in this case).

### Step 4: Substitute values into the formula
Here, \( n = 8 \) and \( r = 4 \). Thus, we need to calculate:
\[ \binom{8 + 4 - 1}{4 - 1} = \binom{11}{3} \]

### Step 5: Calculate \( \binom{11}{3} \)
Using the formula for combinations:
\[ \binom{11}{3} = \frac{11!}{3!(11 - 3)!} = \frac{11!}{3! \cdot 8!} \]
This simplifies to:
\[ \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = \frac{990}{6} = 165 \]

### Conclusion
The number of positive integral solutions to the equation \( a + b + c + d = 12 \) is **165**.

---