Find the coefficient of `x^5` in `(x+3)^6`

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 our case, \( a = x \), \( b = 3 \), and \( n = 6 \). The general term \( T_{r+1} \) in the expansion is given by:

\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r = \binom{6}{r} x^{6-r} (3)^r \]

We need to find the term where the power of \( x \) is 5, which means we need to set \( 6 - r = 5 \).

### Step 1: Determine the value of \( r \)

From the equation \( 6 - r = 5 \):

\[ r = 6 - 5 = 1 \]

### Step 2: Substitute \( r \) back into the general term

Now, we can substitute \( r = 1 \) into the general term:

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

### Step 3: Calculate \( T_{2} \)

Calculating \( \binom{6}{1} \):

\[ \binom{6}{1} = 6 \]

Thus, we have:

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

### Step 4: Identify the coefficient

The coefficient of \( x^5 \) in \( T_{2} \) is:

\[ \text{Coefficient} = 18 \]

### Final Answer

The coefficient of \( x^5 \) in \( (x + 3)^6 \) is **18**.

---