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

Answer

To find the 4th term of the expression \(\left(\frac{3x}{5} - y\right)^{7}\), 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 \(a = \frac{3x}{5}\), \(b = -y\), and \(n = 7\).

### Step 1: Identify the General Term
The general term \(T_{r+1}\) in the expansion is given by:

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

Substituting our values, we get:

\[ T_{r+1} = \binom{7}{r} \left(\frac{3x}{5}\right)^{7-r} (-y)^r \]

### Step 2: Find the 4th Term
To find the 4th term, we need to set \(r = 3\) (since the first term corresponds to \(r = 0\)):

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

### Step 3: Calculate the Binomial Coefficient
Now, we 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: Substitute Values into the Term
Now we substitute back into the term:

\[ T_{4} = 35 \left(\frac{3x}{5}\right)^{4} (-y)^3 \]

### Step 5: Simplify the Expression
Calculating \(\left(\frac{3x}{5}\right)^{4}\):

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

Now, substituting this back into the term:

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

### Step 6: Final Expression
Now, simplifying further:

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

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

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