If the number of integral of solutions of `x+y+z+w lt 25` are `.^(23)C_(lambda)`, such that `x gt -2, y gt 1, z ge 0, w gt 3`, then the value of `lambda` is

Options

A. 3
B. 5
C. 17
D. 19

Answer

To solve the problem of finding the number of integral solutions for the inequality \( x + y + z + w < 25 \) under the given constraints, we will follow these steps:

### Step 1: Rewrite the inequality
We start with the inequality:
\[ x + y + z + w < 25 \]
To convert this into an equation, we introduce a new variable \( \alpha \) such that:
\[ x + y + z + w + \alpha = 25 \]
where \( \alpha > 0 \).

### Step 2: Apply the constraints
The constraints given are:
- \( x > -2 \)
- \( y > 1 \)
- \( z \geq 0 \)
- \( w > 3 \)

To simplify the variables, we can define new variables:
- Let \( x' = x + 2 \) (so \( x' \geq 1 \))
- Let \( y' = y - 2 \) (so \( y' \geq 0 \))
- Let \( z' = z \) (so \( z' \geq 0 \))
- Let \( w' = w - 4 \) (so \( w' \geq 0 \))
- Let \( \alpha' = \alpha - 1 \) (so \( \alpha' \geq 0 \))

### Step 3: Substitute the new variables
Substituting these new variables into the equation gives:
\[ (x' - 2) + (y' + 2) + z' + (w' + 4) + (\alpha' + 1) = 25 \]
This simplifies to:
\[ x' + y' + z' + w' + \alpha' + 5 = 25 \]
Thus, we can rewrite it as:
\[ x' + y' + z' + w' + \alpha' = 20 \]

### Step 4: Count the non-negative integer solutions
Now we need to find the number of non-negative integer solutions to the equation:
\[ x' + y' + z' + w' + \alpha' = 20 \]
The number of solutions to the equation \( a_1 + a_2 + ... + a_k = n \) in non-negative integers is given by the formula:
\[ \binom{n + k - 1}{k - 1} \]
In our case, \( n = 20 \) and \( k = 5 \) (the variables \( x', y', z', w', \alpha' \)). Thus, we have:
\[ \text{Number of solutions} = \binom{20 + 5 - 1}{5 - 1} = \binom{24}{4} \]

### Step 5: Relate to the given expression
The problem states that the number of integral solutions is given as \( \binom{23}{\lambda} \). We need to equate this to our result:
\[ \binom{24}{4} = \binom{23}{\lambda} \]

### Step 6: Use the identity of binomial coefficients
Using the identity \( \binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1} \), we can express \( \binom{24}{4} \) as:
\[ \binom{24}{4} = \binom{23}{4} + \binom{23}{3} \]
This means \( \lambda \) can be either \( 4 \) or \( 3 \).

### Conclusion
Since the problem asks for the value of \( \lambda \), we can conclude:
\[ \lambda = 4 \quad \text{or} \quad \lambda = 3 \]
If we assume \( \lambda \) must be the larger of the two, we can take:
\[ \lambda = 4 \]