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 integral solutions of the equation, such that `(a,b,c,d)lt0`

Options

A. 195
B. -165
C. 92
D. 165

Answer

To find the number of integral solutions for the equation \( a + b + c + d = -12 \) with the condition that \( a, b, c, d < 0 \), we can follow these steps:

### Step 1: Change the Variables
Since \( a, b, c, d < 0 \), we can redefine these variables to make them non-negative. Let:
- \( u = -a \)
- \( v = -b \)
- \( w = -c \)
- \( x = -d \)

Now, since \( a, b, c, d < 0 \), it follows that \( u, v, w, x > 0 \).

### Step 2: Rewrite the Equation
Substituting the new variables into the equation gives:
\[ -u - v - w - x = -12 \]
This simplifies to:
\[ u + v + w + x = 12 \]

### Step 3: Find the Number of Non-Negative Solutions
We need to find the number of integral solutions to the equation \( u + v + w + x = 12 \) where \( u, v, w, x > 0 \). To convert this to a problem involving non-negative integers, we can use the method of "stars and bars".

### Step 4: Adjust for Positive Integers
Since \( u, v, w, x \) must be greater than 0, we can redefine the variables again:
- Let \( u' = u - 1 \)
- Let \( v' = v - 1 \)
- Let \( w' = w - 1 \)
- Let \( x' = x - 1 \)

Now \( u', v', w', x' \geq 0 \) and the equation becomes:
\[ (u' + 1) + (v' + 1) + (w' + 1) + (x' + 1) = 12 \]
This simplifies to:
\[ u' + v' + w' + x' = 8 \]

### Step 5: Apply the Stars and Bars Theorem
Now we need to find the number of non-negative integer solutions to the equation \( u' + v' + w' + x' = 8 \). According to the stars and bars theorem, the number of solutions is given by:
\[ \binom{n + k - 1}{k - 1} \]
where \( n \) is the total number we want (8) and \( k \) is the number of variables (4).

Thus, we have:
\[ \binom{8 + 4 - 1}{4 - 1} = \binom{11}{3} \]

### Step 6: Calculate the Binomial Coefficient
Now we calculate \( \binom{11}{3} \):
\[ \binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = \frac{990}{6} = 165 \]

### Final Answer
The number of integral solutions to the equation \( a + b + c + d = -12 \) with \( a, b, c, d < 0 \) is \( \boxed{165} \).

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