Using binomial theorem, write down the expansions of the following: `(2x+3y)^5`

Explanation

To expand \((2x + 3y)^5\) using the Binomial Theorem, we follow these steps:

### Step 1: Identify the components
The Binomial Theorem states that:
\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]
In our case, \(a = 2x\), \(b = 3y\), and \(n = 5\).

### Step 2: Write the expansion
Using the formula, we can write the expansion as:
\[ (2x + 3y)^5 = \sum_{r=0}^{5} \binom{5}{r} (2x)^{5-r} (3y)^r \]

### Step 3: Calculate each term in the expansion
Now we will calculate each term for \(r = 0\) to \(5\):

1. **For \(r = 0\)**:
\[ \binom{5}{0} (2x)^5 (3y)^0 = 1 \cdot (2x)^5 \cdot 1 = 32x^5 \]

2. **For \(r = 1\)**:
\[ \binom{5}{1} (2x)^4 (3y)^1 = 5 \cdot (16x^4) \cdot (3y) = 240x^4y \]

3. **For \(r = 2\)**:
\[ \binom{5}{2} (2x)^3 (3y)^2 = 10 \cdot (8x^3) \cdot (9y^2) = 720x^3y^2 \]

4. **For \(r = 3\)**:
\[ \binom{5}{3} (2x)^2 (3y)^3 = 10 \cdot (4x^2) \cdot (27y^3) = 1080x^2y^3 \]

5. **For \(r = 4\)**:
\[ \binom{5}{4} (2x)^1 (3y)^4 = 5 \cdot (2x) \cdot (81y^4) = 810xy^4 \]

6. **For \(r = 5\)**:
\[ \binom{5}{5} (2x)^0 (3y)^5 = 1 \cdot 1 \cdot (243y^5) = 243y^5 \]

### Step 4: Combine all terms
Now, we combine all the calculated terms:
\[ (2x + 3y)^5 = 32x^5 + 240x^4y + 720x^3y^2 + 1080x^2y^3 + 810xy^4 + 243y^5 \]

### Final Answer
Thus, the expansion of \((2x + 3y)^5\) is:
\[ 32x^5 + 240x^4y + 720x^3y^2 + 1080x^2y^3 + 810xy^4 + 243y^5 \]