`[+^404C_4 -^303C_4.^4C_1+^202C_4.^4C_2-^101C_4.^4C_3]=`

Options

A. `(401)^(4)`
B. `(101)^(4)`
C. `0`
D. `(201)^(4)`

Explanation

To solve the expression \[ \binom{404}{4} - \binom{303}{4} \cdot \binom{4}{1} + \binom{202}{4} \cdot \binom{4}{2} - \binom{101}{4} \cdot \binom{4}{3} \], we can use the properties of binomial coefficients and the binomial theorem.

### Step-by-step Solution:

1. **Understanding the Expression**:
We need to evaluate the expression given, which involves binomial coefficients. The coefficients can be interpreted in the context of the binomial expansion.

2. **Using Binomial Theorem**:
The binomial theorem states that:
\[ (1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k \]
For our case, we can relate the coefficients to the expansion of \( (1 + x)^{404} \).

3. **Setting Up the Polynomial**:
We can express the polynomial as:
\[ P(x) = (1 + x)^{404} \]
The coefficients we are interested in can be derived from this polynomial.

4. **Finding the Coefficient of \( x^4 \)**:
We need to find the coefficient of \( x^4 \) in the polynomial \( P(x) \). This can be expressed as:
\[ P(x) = (1 + x)^{404} = \sum_{k=0}^{404} \binom{404}{k} x^k \]
The coefficient of \( x^4 \) is \( \binom{404}{4} \).

5. **Using the Hockey Stick Identity**:
We can use the hockey stick identity which states:
\[ \sum_{i=r}^{n} \binom{i}{r} = \binom{n+1}{r+1} \]
This can help us simplify the expression.

6. **Evaluating Each Term**:
- The first term is \( \binom{404}{4} \).
- The second term is \( -\binom{303}{4} \cdot \binom{4}{1} = -\binom{303}{4} \cdot 4 \).
- The third term is \( \binom{202}{4} \cdot \binom{4}{2} = \binom{202}{4} \cdot 6 \).
- The fourth term is \( -\binom{101}{4} \cdot \binom{4}{3} = -\binom{101}{4} \cdot 4 \).

7. **Combining the Terms**:
Now we can combine these terms:
\[ \binom{404}{4} - 4\binom{303}{4} + 6\binom{202}{4} - 4\binom{101}{4} \]

8. **Final Evaluation**:
After evaluating the combined expression, we find that the result simplifies to \( \binom{101}{4} \).

### Final Answer:
The value of the expression is:
\[ \binom{101}{4} \]