Prove that : Find the `5^("th")` term in the expansion of `(3x-4y)^(7)`.

Answer

To find the 5th term in the expansion of \((3x - 4y)^7\), 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, we have \(a = 3x\), \(b = -4y\), and \(n = 7\).

### Step 1: Identify the term we need
We want to find the 5th term in the expansion. In the Binomial expansion, the \( (r+1) \)th term is given by:

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

To find the 5th term, we set \(r + 1 = 5\), which gives us \(r = 4\).

### Step 2: Substitute values into the formula
Now we can substitute \(n = 7\), \(r = 4\), \(a = 3x\), and \(b = -4y\) into the formula:

\[ T_5 = \binom{7}{4} (3x)^{7-4} (-4y)^4 \]

### Step 3: Calculate each component
1. Calculate \(\binom{7}{4}\):

\[ \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

2. Calculate \((3x)^{3}\):

\[ (3x)^{3} = 27x^{3} \]

3. Calculate \((-4y)^{4}\):

\[ (-4y)^{4} = 256y^{4} \]

### Step 4: Combine all parts
Now we can combine all the parts together:

\[ T_5 = 35 \times 27x^{3} \times 256y^{4} \]

### Step 5: Calculate the final coefficient
Now, we calculate the coefficient:

\[ 35 \times 27 \times 256 \]

Calculating this step-by-step:

- First, calculate \(35 \times 27 = 945\)
- Then, calculate \(945 \times 256 = 241920\)

### Step 6: Write the final term
Thus, the 5th term is:

\[ T_5 = 241920 x^{3} y^{4} \]

### Final Answer:
The 5th term in the expansion of \((3x - 4y)^{7}\) is:

\[ 241920 x^{3} y^{4} \]

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