In a sample of hydrogen atoms, transition of electrons take place from the $5^{th}$ excited state to the ground state producing all possible types of photons. The number of lines in the infrared region is:

The correct option is C $6$
All the possible transitions emit radiation either in the ultraviolet, visible or infrared band.
So, the number of lines in the infrared region = total emitted lines $-($Number of lines in the visible region $+$ Number of lines in the ultraviolet region)
We know that, $n=6$ for the $5^{th}$ excited state.
Total emitted lines for transition, $n_1=6\to n_2=1$$=\frac{(n_1-n_2)(n_1-n_2+1)}{2}=15$

Since all the lines falling onto $n=2$ are in the visible band, the number of lines in the visible region = $4(transitionsfrom6\to 2,5\to 2,4\to 2,3\to 2)$

Since all the lines falling onto $n=1$ are in the ultraviolet band, the number of lines in the ultraviolet region =$5(transitionsfrom6\to 1,5\to 1,4\to 1,3\to 1,2\to 1)$

Now, No. of lines in Infrared region = $15-4-5=6$