Among the following series of transition metal ions, how many metals ions have `d^(2)` electronic configuration
`Fe^(4+), V^(3+), Sn^(2+), Au^(3+)Ti^(2+), As^(3+), Cr^(3+), Mn^(5+)Co^(3+), Zr^(2+), Cd^(2+), Mo^(4+), La^(2+), Ta^(4+), Co^(6+), Ni^(4+)`

Answer

To determine how many transition metal ions from the given list have a `d^(2)` electronic configuration, we will analyze each ion step by step.

### Step-by-Step Solution:

1. **Fe^(4+)**:
- Atomic number of Fe = 26.
- Electronic configuration of Fe: [Ar] 4s² 3d⁶.
- For Fe^(4+), remove 4 electrons (first from 4s, then from 3d):
- Configuration: [Ar] 4s⁰ 3d⁴ (not d²).

2. **V^(3+)**:
- Atomic number of V = 23.
- Electronic configuration of V: [Ar] 4s² 3d³.
- For V^(3+), remove 3 electrons:
- Configuration: [Ar] 4s⁰ 3d² (this is d²).

3. **Sn^(2+)**:
- Atomic number of Sn = 50.
- Electronic configuration of Sn: [Kr] 5s² 4d¹⁰ 5p².
- For Sn^(2+), remove 2 electrons:
- Configuration: [Kr] 5s⁰ 4d¹⁰ 5p² (not d²).

4. **Au^(3+)**:
- Atomic number of Au = 79.
- Electronic configuration of Au: [Xe] 4f¹⁴ 5d¹⁰ 6s¹.
- For Au^(3+), remove 3 electrons:
- Configuration: [Xe] 4f¹⁴ 5d⁸ 6s⁰ (not d²).

5. **Ti^(2+)**:
- Atomic number of Ti = 22.
- Electronic configuration of Ti: [Ar] 4s² 3d².
- For Ti^(2+), remove 2 electrons:
- Configuration: [Ar] 4s⁰ 3d² (this is d²).

6. **As^(3+)**:
- Atomic number of As = 33.
- Electronic configuration of As: [Ar] 4s² 3d¹⁰ 4p³.
- For As^(3+), remove 3 electrons:
- Configuration: [Ar] 4s² 3d¹⁰ 4p⁰ (not d²).

7. **Cr^(3+)**:
- Atomic number of Cr = 24.
- Electronic configuration of Cr: [Ar] 4s¹ 3d⁵.
- For Cr^(3+), remove 3 electrons:
- Configuration: [Ar] 4s⁰ 3d³ (not d²).

8. **Mn^(5+)**:
- Atomic number of Mn = 25.
- Electronic configuration of Mn: [Ar] 4s² 3d⁵.
- For Mn^(5+), remove 5 electrons:
- Configuration: [Ar] 4s⁰ 3d² (this is d²).

9. **Co^(3+)**:
- Atomic number of Co = 27.
- Electronic configuration of Co: [Ar] 4s² 3d⁷.
- For Co^(3+), remove 3 electrons:
- Configuration: [Ar] 4s⁰ 3d⁶ (not d²).

10. **Zr^(2+)**:
- Atomic number of Zr = 40.
- Electronic configuration of Zr: [Kr] 5s² 4d².
- For Zr^(2+), remove 2 electrons:
- Configuration: [Kr] 5s⁰ 4d² (this is d²).

11. **Cd^(2+)**:
- Atomic number of Cd = 48.
- Electronic configuration of Cd: [Kr] 5s² 4d¹⁰.
- For Cd^(2+), remove 2 electrons:
- Configuration: [Kr] 5s⁰ 4d¹⁰ (not d²).

12. **Mo^(4+)**:
- Atomic number of Mo = 42.
- Electronic configuration of Mo: [Kr] 5s¹ 4d⁵.
- For Mo^(4+), remove 4 electrons:
- Configuration: [Kr] 5s⁰ 4d² (this is d²).

13. **La^(2+)**:
- Atomic number of La = 57.
- Electronic configuration of La: [Xe] 6s² 5d¹.
- For La^(2+), remove 2 electrons:
- Configuration: [Xe] 6s⁰ 5d¹ (not d²).

14. **Ta^(4+)**:
- Atomic number of Ta = 73.
- Electronic configuration of Ta: [Xe] 4f¹⁴ 5d³ 6s².
- For Ta^(4+), remove 4 electrons:
- Configuration: [Xe] 4f¹⁴ 5d² 6s⁰ (this is d²).

15. **Co^(6+)**:
- Atomic number of Co = 27.
- Electronic configuration of Co: [Ar] 4s² 3d⁷.
- For Co^(6+), remove 6 electrons:
- Configuration: [Ar] 4s⁰ 3d¹ (not d²).

16. **Ni^(4+)**:
- Atomic number of Ni = 28.
- Electronic configuration of Ni: [Ar] 4s² 3d⁸.
- For Ni^(4+), remove 4 electrons:
- Configuration: [Ar] 4s⁰ 3d⁶ (not d²).

### Summary of Results:
The ions with `d²` electronic configuration are:
- V^(3+)
- Ti^(2+)
- Mn^(5+)
- Zr^(2+)
- Mo^(4+)
- Ta^(4+)

Thus, the total number of transition metal ions with a `d²` electronic configuration is **6**.