Question

The coefficient of $x^{9}$ in the expansion of $(4-\frac{x^{2}}{4})^{12}$ is ________.

• A. $-^{12}C_{7}(4)^{7}(3)^{5}$
• B. $^{12}C_{7}(4)^{7}(3)^{5}$
• C. $^{12}C_{6}(4)^{6}(3)^{6}$
• D. $^{12}C_{5}(4)^{5}(3)^{7}$
• E. 0

Hint:

If only $x^2$ is present, all odd power coefficients are zero.

Answer 1

Answer: (E)

Step 1: Understanding the Concept
We need to find the coefficient of a specific power of \(x\) in a binomial expansion. The key is to analyze the general term of the expansion and see which powers of \(x\) can be generated.
Step 2: Key Formula or Approach
The general term (the \((r+1)\)-th term) in the binomial expansion of \((A+B)^n\) is given by the formula:
\[ T_{r+1} = {}^nC_r A^{n-r} B^r \] We will apply this formula to our expression and inspect the resulting power of \(x\).
Step 3: Detailed Explanation
1. Identify A, B, and n.
For the expansion of \((4 - \frac{x^2}{3})^{12}\):
\(A = 4\)
\(B = -\frac{x^2}{3}\)
\(n = 12\)
2. Write down the general term.
\[ T_{r+1} = {}^{12}C_r (4)^{12-r} \left(-\frac{x^2}{3}\right)^r \] 3. Separate the parts of the term.
We can separate the constant part, the sign, and the variable part.
\[ T_{r+1} = {}^{12}C_r (4)^{12-r} \frac{(-1)^r}{3^r} (x^2)^r \] 4. Determine the power of x.
The part of the term involving \(x\) is \((x^2)^r\). Using the law of exponents \((x^a)^b = x^{ab}\), this simplifies to:
\[ x^{2r} \] 5. Analyze the possible powers of x.
The value of \(r\) in a binomial expansion can be any integer from 0 to \(n\). In this case, \(r\) can be \(0, 1, 2, \dots, 12\).
The power of \(x\) in any term is always \(2r\). This means the possible powers of \(x\) in the expansion are:
\(2 \times 0 = 0\) (for \(x^0\))
\(2 \times 1 = 2\) (for \(x^2\))
\(2 \times 2 = 4\) (for \(x^4\))
...
\(2 \times 12 = 24\) (for \(x^{24}\))
All the powers of \(x\) in the expansion are even integers.
6. Conclude about the coefficient of \(x^7\).
The question asks for the coefficient of \(x^7\). Since 7 is an odd number, it is impossible to generate a term with \(x^7\) from this expansion. Therefore, the coefficient of the \(x^7\) term must be 0.
Step 4: Final Answer
The coefficient of \(x^7\) is 0.