`15^(th)` term of `(2x-3y)^(20)`

Answer

To find the 15th term of the expression \((2x - 3y)^{20}\), 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, \(a = 2x\), \(b = -3y\), and \(n = 20\).

### Step-by-Step Solution:

1. **Identify the term we need**: We want the 15th term. In the binomial expansion, the \(r\)th term is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
Therefore, the 15th term corresponds to \(r = 14\) (since we start counting from \(r = 0\)).

2. **Substitute values into the formula**:
\[ T_{15} = \binom{20}{14} (2x)^{20-14} (-3y)^{14} \]

3. **Calculate \(n - r\) and \(r\)**:
- \(n - r = 20 - 14 = 6\)
- \(r = 14\)

4. **Write the term explicitly**:
\[ T_{15} = \binom{20}{14} (2x)^6 (-3y)^{14} \]

5. **Calculate the binomial coefficient**:
\[ \binom{20}{14} = \binom{20}{6} \quad \text{(using the property } \binom{n}{r} = \binom{n}{n-r}\text{)} \]

6. **Calculate \((2x)^6\)**:
\[ (2x)^6 = 2^6 x^6 = 64x^6 \]

7. **Calculate \((-3y)^{14}\)**:
\[ (-3y)^{14} = (-1)^{14} 3^{14} y^{14} = 3^{14} y^{14} \quad \text{(since } (-1)^{14} = 1\text{)} \]

8. **Combine all parts**:
\[ T_{15} = \binom{20}{14} \cdot 64x^6 \cdot 3^{14} y^{14} \]

9. **Final expression**:
\[ T_{15} = \binom{20}{14} \cdot 64 \cdot 3^{14} \cdot x^6 \cdot y^{14} \]

### Final Answer:
The 15th term of \((2x - 3y)^{20}\) is:
\[ T_{15} = \binom{20}{14} \cdot 64 \cdot 3^{14} \cdot x^6 \cdot y^{14} \]