How many of the following options are incorrect in accordance with the mentioned properties?
(I) `IP_1` of ion `M^(2+) gt EA1` of `M^(3+)`.(IP = ionization potential, and EA = electron affinity)
(II) `S lt Se lt Te lt O`(order of EA)
(III) `Li lt Be lt B  lt C`(order of electronegativity)
(IV) `Mn^(4+) lt Mg^(2+) lt Na^(+) lt K^(+)` (order of ionic size)
(V) `Li^(+) gt Na^(+) lt K^(+)` (order of hydrated size)

Answer

To determine how many of the given options are incorrect based on the mentioned properties, we will analyze each statement one by one.

### Step-by-Step Solution:

1. **Statement I: `IP_1` of ion `M^(2+)` > `EA_1` of `M^(3+)`**
- **Analysis**: Ionization potential (IP) refers to the energy required to remove an electron from an atom or ion. Electron affinity (EA) refers to the energy released when an electron is added to an atom or ion.
- For `M^(2+)`, removing an electron to form `M^(3+)` requires energy, hence IP will be relatively high. For `M^(3+)`, adding an electron releases energy, but since `M^(2+)` has a higher nuclear charge (due to losing two electrons), it will require more energy to remove an electron compared to the energy released when adding an electron to `M^(3+)`.
- **Conclusion**: This statement is **correct**.

2. **Statement II: `S < Se < Te < O` (order of EA)**
- **Analysis**: As we move down the group in the periodic table, the electron affinity generally decreases because the atomic size increases, leading to a decrease in the attraction between the nucleus and the added electron. However, oxygen has a very small size which leads to a significant repulsion when adding an electron, resulting in a lower electron affinity compared to sulfur, selenium, and tellurium.
- The correct order should be `O < S < Se < Te`.
- **Conclusion**: This statement is **incorrect**.

3. **Statement III: `Li < Be < B < C` (order of electronegativity)**
- **Analysis**: Electronegativity increases across a period from left to right due to increasing nuclear charge. Therefore, the correct order of electronegativity should be `C > B > Be > Li`.
- **Conclusion**: This statement is **incorrect**.

4. **Statement IV: `Mn^(4+) < Mg^(2+) < Na^(+) < K^(+)` (order of ionic size)**
- **Analysis**: The ionic size decreases with increasing positive charge. Therefore, `Mg^(2+)` should be smaller than `Na^(+)`, and `Na^(+)` should be smaller than `K^(+)`. The order should be `Mn^(4+) < Mg^(2+) < Na^(+) < K^(+)` as `K^(+)` is the largest due to having more shells.
- **Conclusion**: This statement is **incorrect**.

5. **Statement V: `Li^(+) > Na^(+) < K^(+)` (order of hydrated size)**
- **Analysis**: The hydrated size is inversely proportional to the ionic charge density. As we move down the group, the size of the ions increases. Therefore, the correct order should be `K^(+) > Na^(+) > Li^(+)` since `K^(+)` is the largest and `Li^(+)` is the smallest.
- **Conclusion**: This statement is **incorrect**.

### Summary of Incorrect Statements:
- Statement II: Incorrect
- Statement III: Incorrect
- Statement IV: Incorrect
- Statement V: Incorrect

Thus, **4 out of the 5 statements are incorrect**.

### Final Answer:
**4 statements are incorrect.**