According to the question, the electron in a hydrogen-like sample makes a transition from the 4th excited state to the 2nd state. We need to determine the number of spectral lines observed and the number of lines belonging to the Balmer and Paschen series.

Explanation:

Transition between energy levels:
- In a hydrogen-like sample, the energy levels of the electron are given by the formula: E = -13.6/n^2, where n is the principal quantum number.
- The 4th excited state has a principal quantum number (n) of 4, and the 2nd state has a principal quantum number of 2.
- By substituting these values into the formula, we find that the energy difference between the two states is ΔE = E2 - E4 = -13.6/2^2 - (-13.6/4^2) = 10.2 eV.

Number of spectral lines:
- The number of spectral lines observed is equal to the number of possible electronic transitions between the two energy levels.
- The number of possible transitions can be calculated using the formula: ΔE = hc/λ, where ΔE is the energy difference, h is Planck's constant, c is the speed of light, and λ is the wavelength of the spectral line.
- Rearranging the formula, we get: λ = hc/ΔE.
- Substituting the values, we find: λ = (6.626 × 10^-34 J s × 3 × 10^8 m/s) / (10.2 eV × 1.6 × 10^-19 J/eV) ≈ 3.07 × 10^-7 m.
- This wavelength corresponds to the visible light spectrum, and there are typically 6 different spectral lines observed in this range.

Number of lines belonging to the Balmer series:
- The Balmer series consists of spectral lines corresponding to transitions from higher energy levels to the second energy level (n = 2).
- In this case, the electron is transitioning from the 4th excited state to the 2nd state, which is a transition within the Balmer series.
- Therefore, the number of lines belonging to the Balmer series is 3.

Number of lines belonging to the Paschen series:
- The Paschen series consists of spectral lines corresponding to transitions from higher energy levels to the third energy level (n = 3).
- In this case, the electron is transitioning from the 4th excited state to the 2nd state, which is not a transition within the Paschen series.
- Therefore, the number of lines belonging to the Paschen series is 0.

Conclusion:
Based on the calculations and the given information, the correct answer is option 'B, C, D'. There are 6 different spectral lines observed, 3 of which belong to the Balmer series and 0 belong to the Paschen series.