Find the third term in the expansion of
`(x + 2/5 y)^4`

Answer

To find the third term in the expansion of \((x + \frac{2}{5}y)^4\), we can use the Binomial Theorem. The Binomial Theorem states that:

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

where \(\binom{n}{r}\) is the binomial coefficient, \(a\) and \(b\) are the terms being expanded, and \(n\) is the exponent.

### Step-by-Step Solution:

1. **Identify the values**:
- Here, \(a = x\), \(b = \frac{2}{5}y\), and \(n = 4\).

2. **Determine the term we need**:
- We are looking for the third term, which corresponds to \(T_{r+1}\) where \(r = 2\) (since the first term corresponds to \(r = 0\)).

3. **Use the formula for 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 \(n = 4\) and \(r = 2\):
\[ T_3 = \binom{4}{2} x^{4-2} \left(\frac{2}{5}y\right)^2 \]

4. **Calculate the binomial coefficient**:
- Calculate \(\binom{4}{2}\):
\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

5. **Substitute back into the term**:
- Now substituting the values back into the term:
\[ T_3 = 6 \cdot x^{2} \cdot \left(\frac{2}{5}y\right)^2 \]

6. **Calculate \(\left(\frac{2}{5}y\right)^2\)**:
- This gives:
\[ \left(\frac{2}{5}y\right)^2 = \frac{4}{25}y^2 \]

7. **Combine the results**:
- Now substituting this back into \(T_3\):
\[ T_3 = 6 \cdot x^{2} \cdot \frac{4}{25}y^2 = \frac{24}{25}x^2y^2 \]

### Final Answer:
The third term in the expansion of \((x + \frac{2}{5}y)^4\) is:
\[ \frac{24}{25}x^2y^2 \]