`4^(th)` terms of `((3x)/(5)-y)^(7)`

Answer

To find the fourth term of the expression \(\left(\frac{3x}{5} - y\right)^{7}\), we can use the Binomial Theorem, which states that:

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

In this case, we have:
- \(a = \frac{3x}{5}\)
- \(b = -y\)
- \(n = 7\)

### Step 1: Identify the term we need
We need to find the fourth term in the expansion. The general term (the \((k+1)^{th}\) term) in the expansion is given by:

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

To find the fourth term, we set \(k = 3\) (since we start counting from \(k = 0\)).

### Step 2: Substitute values into the formula
Now we substitute \(n = 7\), \(k = 3\), \(a = \frac{3x}{5}\), and \(b = -y\) into the formula:

\[ T_{4} = \binom{7}{3} \left(\frac{3x}{5}\right)^{7-3} (-y)^3 \]

### Step 3: Calculate the binomial coefficient
Calculate \(\binom{7}{3}\):

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

### Step 4: Calculate the powers
Now, calculate the powers:

\[ \left(\frac{3x}{5}\right)^{4} = \frac{(3x)^4}{5^4} = \frac{81x^4}{625} \]

And for \((-y)^3\):

\[ (-y)^3 = -y^3 \]

### Step 5: Combine all parts
Now, we can combine all parts to get the fourth term:

\[ T_{4} = 35 \cdot \frac{81x^4}{625} \cdot (-y^3) = -\frac{2835x^4y^3}{625} \]

### Final Answer
Thus, the fourth term of \(\left(\frac{3x}{5} - y\right)^{7}\) is:

\[ -\frac{2835x^4y^3}{625} \]

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