The number of integral solutions for the equation
a+b+c+d = 12, where `(a,b,c,d) ge -1` is :

Options

A. `"^(19)C_3`
B. `"^(18)C_4`
C. `"^(20)C_4`
D. none of these

Answer

To solve the equation \( a + b + c + d = 12 \) where \( a, b, c, d \geq -1 \), we can follow these steps:

### Step 1: Transform the variables
Since \( a, b, c, d \) can take values starting from \(-1\), we can redefine the variables to make them non-negative. Let:
- \( a' = a + 1 \)
- \( b' = b + 1 \)
- \( c' = c + 1 \)
- \( d' = d + 1 \)

This transformation ensures that \( a', b', c', d' \geq 0 \).

### Step 2: Rewrite the equation
Substituting the new variables into the original equation gives us:
\[ (a' - 1) + (b' - 1) + (c' - 1) + (d' - 1) = 12 \]
This simplifies to:
\[ a' + b' + c' + d' - 4 = 12 \]
or
\[ a' + b' + c' + d' = 16 \]

### Step 3: Count the non-negative integer solutions
Now we need to find the number of non-negative integer solutions to the equation \( a' + b' + c' + d' = 16 \).

The formula for the number of non-negative integer solutions to the equation \( x_1 + x_2 + ... + x_r = n \) is given by:
\[ \binom{n + r - 1}{r - 1} \]
where \( n \) is the total sum we want (16 in this case) and \( r \) is the number of variables (4 in this case: \( a', b', c', d' \)).

### Step 4: Apply the formula
Here, \( n = 16 \) and \( r = 4 \). Therefore, we calculate:
\[ \binom{16 + 4 - 1}{4 - 1} = \binom{19}{3} \]

### Step 5: Calculate \( \binom{19}{3} \)
Now we compute \( \binom{19}{3} \):
\[ \binom{19}{3} = \frac{19 \times 18 \times 17}{3 \times 2 \times 1} = \frac{5814}{6} = 969 \]

### Final Answer
Thus, the number of integral solutions for the equation \( a + b + c + d = 12 \) where \( a, b, c, d \geq -1 \) is **969**.

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