The number of integral solutions of the inequation `x+y+z le 100, (x ge 2, y ge 3, z ge 4)` , is

Options

A. `.^(100)C_(2)`
B. `.^(94)C_(3)`
C. `.^(93)C_(2)`
D. None of these

Answer

To find the number of integral solutions for the inequality \( x + y + z \leq 100 \) with the constraints \( x \geq 2 \), \( y \geq 3 \), and \( z \geq 4 \), we can follow these steps:

### Step 1: Change of Variables
We will substitute variables to simplify the constraints:
- Let \( x = t + 2 \) where \( t \geq 0 \)
- Let \( y = m + 3 \) where \( m \geq 0 \)
- Let \( z = n + 4 \) where \( n \geq 0 \)

### Step 2: Substitute in the Inequality
Substituting these new variables into the inequality gives us:
\[ (t + 2) + (m + 3) + (n + 4) \leq 100 \]
This simplifies to:
\[ t + m + n + 9 \leq 100 \]

### Step 3: Rearranging the Inequality
Rearranging the inequality, we get:
\[ t + m + n \leq 100 - 9 \]
\[ t + m + n \leq 91 \]

### Step 4: Counting Non-Negative Solutions
Now, we need to count the number of non-negative integer solutions to the equation:
\[ t + m + n + k = 91 \]
where \( k \) is a non-negative integer that accounts for the "less than or equal to" part. Here, \( k \) can take values from \( 0 \) to \( 91 \).

### Step 5: Using Stars and Bars Theorem
The number of non-negative integer solutions to the equation \( t + m + n + k = 91 \) can be found using the stars and bars theorem. The number of solutions is given by:
\[ \binom{n + r - 1}{r - 1} \]
where \( n \) is the total number (91) and \( r \) is the number of variables (4: \( t, m, n, k \)).

Thus, we have:
\[ \text{Number of solutions} = \binom{91 + 4 - 1}{4 - 1} = \binom{94}{3} \]

### Conclusion
Therefore, the total number of integral solutions of the inequality \( x + y + z \leq 100 \) with the given constraints is:
\[ \boxed{94C3} \]