The number of integers solutions for the equation `x+y+z+t=20`, where x,y,z t are all `ge-1`, is

Options

A. `""^(20)C_(4)`
B. `""^(23)C_(3)`
C. `""^(27)C_(4)`
D. `""^(27)C_(3)`

Explanation

To find the number of integer solutions for the equation \( x + y + z + t = 20 \) where \( x, y, z, t \geq -1 \), we can follow these steps:

### Step 1: Transform the Variables
Since \( x, y, z, t \) are all greater than or equal to -1, we can redefine these variables to make them non-negative. We can set:
- \( x' = x + 1 \)
- \( y' = y + 1 \)
- \( z' = z + 1 \)
- \( t' = t + 1 \)

This transformation implies that \( x', y', z', t' \geq 0 \).

### Step 2: Rewrite the Equation
Substituting the new variables into the original equation gives us:
\[ (x' - 1) + (y' - 1) + (z' - 1) + (t' - 1) = 20 \]
This simplifies to:
\[ x' + y' + z' + t' - 4 = 20 \]
Thus, we can rewrite it as:
\[ x' + y' + z' + t' = 24 \]

### Step 3: Count the Non-Negative Integer Solutions
Now, we need to find the number of non-negative integer solutions to the equation \( x' + y' + z' + t' = 24 \). This is a classic problem that can be solved using the "stars and bars" theorem.

According to the stars and bars theorem, the number of solutions to the equation \( x_1 + x_2 + ... + x_r = n \) in non-negative integers is given by:
\[ \binom{n + r - 1}{r - 1} \]
where \( n \) is the total number (24 in this case) and \( r \) is the number of variables (4 in this case).

### Step 4: Apply the Formula
Here, \( n = 24 \) and \( r = 4 \). Thus, we have:
\[ \text{Number of solutions} = \binom{24 + 4 - 1}{4 - 1} = \binom{27}{3} \]

### Conclusion
The number of integer solutions for the equation \( x + y + z + t = 20 \) where \( x, y, z, t \geq -1 \) is:
\[ \binom{27}{3} \]