Given that, Ramesh score $ = 82 \cdot 5 $
And, the average score of the $21$ students other than Gautam is $62.$
$\frac{\text{Ramesh + 20 others (not include Gautam)}} {21} = 62 $
$\Rightarrow 20$ others students score $ = 62 \times 21 – 82 \cdot 5 = 1302 – 82 \cdot 5 = 1219 \cdot 5$
The average score of all the $22$ students is one more than the average score of the $21$ students other than Ramesh.
$\frac{ \text {Ramesh + Gautam +20 others}} {22} = \frac{ \text {Gautam + 20 others (Ramesh not include)}}{21} + 1 $
$\Rightarrow \frac{ 82\cdot5 + \text{Gautam} + 1219\cdot5 }{22} = \frac{ \text{Gautam} + 1219\cdot5 }{21} + 1 $
$ \Rightarrow \frac{1302 + \text{Gautam}}{22} = \frac{\text{Gautam} + 1219\cdot5 + 21}{21} $
$ \Rightarrow \frac{1302 + \text{Gautam}}{22} = \frac{\text{Gautam} + 1240\cdot5}{21} $
$ \Rightarrow 1302 \times 21 + 21 \; \text{Gautam} = 22 \; \text{Gautam} + 1240 \cdot5 \times 22 $
$ \Rightarrow 27342 + 21 \; \text{Gautam} = 22 \; \text{Gautam} + 27291 $
$ \Rightarrow 27342 – 27291 = 22 \; \text{Gautam} – 21 \; \text{Gautam} $
$ \Rightarrow \boxed{\text{Gautam} = 51} $
$\therefore$ The score of Gautam $ = 51.$
Correct Answer $: \text{C}$