If the value of `"^(n)C_(0)+2*^(n)C_(1)+3*^(n)C_(2)+...+(n+1)*^(n)C_(n)=576`, then `n` is (a) 7 (b) 5 (c) 6 (d) 9

Options

A. `7`
B. `5`
C. `6`
D. `9`

Explanation

To solve the problem, we need to evaluate the expression given and find the value of \( n \) such that:

\[ ^{n}C_{0} + 2 \cdot ^{n}C_{1} + 3 \cdot ^{n}C_{2} + \ldots + (n+1) \cdot ^{n}C_{n} = 576 \]

### Step 1: Understand the expression
The expression can be interpreted using the Binomial Theorem. The term \( k \cdot ^{n}C_{k-1} \) can be derived from differentiating the binomial expansion.

### Step 2: Use the Binomial Theorem
Recall that:
\[ (1 + x)^n = \sum_{k=0}^{n} ^{n}C_{k} x^k \]

### Step 3: Multiply by \( x \)
Multiplying both sides by \( x \):
\[ x(1 + x)^n = \sum_{k=0}^{n} ^{n}C_{k} x^{k+1} \]

### Step 4: Differentiate both sides
Now differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}[x(1 + x)^n] = (1 + x)^n + nx(1 + x)^{n-1} \]
The right side becomes:
\[ \sum_{k=0}^{n} k \cdot ^{n}C_{k} x^{k} \]

### Step 5: Set \( x = 1 \)
Setting \( x = 1 \):
\[ (1 + 1)^n + n(1 + 1)^{n-1} = \sum_{k=0}^{n} k \cdot ^{n}C_{k} \]
This simplifies to:
\[ 2^n + n \cdot 2^{n-1} \]

### Step 6: Equate to 576
Now we have:
\[ 2^n + n \cdot 2^{n-1} = 576 \]

### Step 7: Factor out \( 2^{n-1} \)
Factoring out \( 2^{n-1} \):
\[ 2^{n-1}(2 + n) = 576 \]

### Step 8: Solve for \( n \)
Dividing both sides by \( 2^{n-1} \):
\[ 2 + n = \frac{576}{2^{n-1}} \]

### Step 9: Test possible values of \( n \)
We can test the options given in the question:

1. **For \( n = 5 \)**:
\[ 2 + 5 = 7 \quad \Rightarrow \quad 2^{4} = 16 \quad \Rightarrow \quad 7 \cdot 16 = 112 \quad \text{(not equal to 576)} \]

2. **For \( n = 6 \)**:
\[ 2 + 6 = 8 \quad \Rightarrow \quad 2^{5} = 32 \quad \Rightarrow \quad 8 \cdot 32 = 256 \quad \text{(not equal to 576)} \]

3. **For \( n = 7 \)**:
\[ 2 + 7 = 9 \quad \Rightarrow \quad 2^{6} = 64 \quad \Rightarrow \quad 9 \cdot 64 = 576 \quad \text{(correct)} \]

4. **For \( n = 9 \)**:
\[ 2 + 9 = 11 \quad \Rightarrow \quad 2^{8} = 256 \quad \Rightarrow \quad 11 \cdot 256 = 2816 \quad \text{(not equal to 576)} \]

### Conclusion
The value of \( n \) that satisfies the equation is:
\[ \boxed{7} \]