The coefficient of `x^(5)` in the expansion of `(x +3)^(6)`,is

Options

A. 18
B. 6
C. 12
D. 10

Answer

To find the coefficient of \( x^5 \) in the expansion of \( (x + 3)^6 \), we can use the Binomial Theorem. The Binomial Theorem states that:

\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]

In this case, we have:
- \( a = x \)
- \( b = 3 \)
- \( n = 6 \)

### Step 1: Identify the General Term
The general term \( T_{r+1} \) in the expansion is given by:

\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]

Substituting the values of \( a \), \( b \), and \( n \):

\[ T_{r+1} = \binom{6}{r} x^{6-r} \cdot 3^r \]

### Step 2: Find the Value of \( r \) for \( x^5 \)
We need to find the term where the power of \( x \) is 5. This occurs when:

\[ 6 - r = 5 \]

Solving for \( r \):

\[ r = 6 - 5 = 1 \]

### Step 3: Substitute \( r \) into the General Term
Now we substitute \( r = 1 \) back into the general term:

\[ T_{2} = \binom{6}{1} x^{6-1} \cdot 3^1 \]

### Step 4: Calculate the Coefficient
Calculating \( T_{2} \):

\[ T_{2} = \binom{6}{1} x^{5} \cdot 3 = 6 \cdot x^{5} \cdot 3 = 18 x^{5} \]

### Step 5: Identify the Coefficient of \( x^5 \)
The coefficient of \( x^5 \) in the expansion is:

\[ \text{Coefficient} = 18 \]

Thus, the coefficient of \( x^5 \) in the expansion of \( (x + 3)^6 \) is **18**.

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