Explanation:

To determine the number of lines in the infrared region when an electron in the hydrogen atom makes a transition from the 5th excited state to the ground state, we need to consider the energy levels involved and the possible transitions.

Energy Levels:
In the hydrogen atom, the energy levels are given by the formula:
E = -13.6/n^2 eV
where n is the principal quantum number.

For the 5th excited state (n = 5), the energy would be:
E5 = -13.6/5^2 = -0.544 eV

For the ground state (n = 1), the energy would be:
E1 = -13.6/1^2 = -13.6 eV

Transition and Energy Difference:
The transition from the 5th excited state to the ground state involves an electron moving from a higher energy level to a lower energy level, resulting in the emission of a photon.

The energy difference between the two levels can be calculated as:
ΔE = E1 - E5 = -13.6 - (-0.544) = -13.056 eV

Photon Energy:
The energy of a photon is given by the equation:
E = hc/λ
where E is the energy of the photon, h is Planck's constant, c is the speed of light, and λ is the wavelength of the photon.

In the infrared region, the wavelength of the photon would typically be greater than 700 nm.

Calculating Possible Wavelengths:
To determine the possible wavelengths of the emitted photons, we can use the equation:
ΔE = hc/λ

Solving for λ, we get:
λ = hc/ΔE

Substituting the values, we get:
λ = (6.63 x 10^-34 J s * 3 x 10^8 m/s) / (13.056 eV * 1.6 x 10^-19 J/eV)

Calculating this, we find that the wavelength is approximately 9.59 x 10^-6 m, which corresponds to the infrared region.

Number of Lines:
Since the question asks for the number of lines in the infrared region, we need to consider the possible transitions.

In this case, the electron is moving from the 5th excited state to the ground state. Since the energy levels are discrete, the electron can only make one transition.

Therefore, there is only one line in the infrared region.

Hence, the correct answer is option B) 1 line.