$\text{There are 10 attendees in total,}\\ \text{Shyamal counts all the hands shaken by everyone other than him.}$
$\text{no. of hands that other attendees including his spouse shook (expcept him) }$
$\quad \quad= \sum_{i=0}^{8}(i) = \frac{8\times 9}{2} = 36.$
$\text{Hand shaking lemma state that,} \\ \text{The sum of total hand shakes(sum of degrees) should be even,} \\ \text{total no. of handshakes expect him is 36, }$
$\text{Therefore, No of persons who shook hands with Shyamal should be even.} \\ \quad \quad \text{(Let it be n)}$
$\text{There are 5 different possible values of n \{0,2,4,6,8\}, } \\ \text{and 5 different degree sequences as follows:}$
$ n =8, \quad \\ \quad \text{degree sequence: } \\ \quad \quad 8,8,7,6,5,4,3,2,1,0 \\ \quad \quad 0,7,6,5,4,3,2,1,0,0 \quad \quad \rightarrow \text{Not possible, we can’t delete 7 now.}$
$ n =6, \quad \\ \quad \text{degree sequence: } \\ \quad \quad 8,7,6,6,5,4,3,2,1,0 \\ \quad \quad 0,6,5,5,4,3,2,1,0,0 \\ \quad \quad 0,0,4,4,3,2,1,0,0,0 \\ \quad \quad 0,0,0,3,2,1,0,0,0,0 \quad \quad \rightarrow \text{Not possible, we can’t delete 3 now.}$
$ n =4, \quad \\ \quad \text{degree sequence: } \\ \quad \quad 8,7,6,5,4,4,3,2,1,0 \\ \quad \quad 0,6,5,4,3,3,2,1,0,0 \\ \quad \quad 0,0,4,3,2,2,1,0,0,0 \\ \quad \quad 0,0,0,2,1,1,0,0,0,0 \\ \quad \quad 0,0,0,0,0,0,0,0,0,0 \quad \quad \rightarrow \textbf{one possible degree sequence.}$
$ n =2, \quad \\ \quad \text{degree sequence: } \\ \quad \quad 8,7,6,5,4,3,2,2,1,0 \\ \quad \quad 0,6,5,4,3,2,1,1,0,0 \\ \quad \quad 0,0,4,3,2,1,0,0,0,0 \quad \quad \rightarrow \text{Not possible, we can’t delete 4 now.}$
$ n =0, \quad \\ \quad \text{degree sequence: } \\ \quad \quad 8,7,6,5,4,3,2,1,0,0 \quad \quad \rightarrow \text{Not possible, we can’t even delete 8}$
$\text{Only possible value for n is 4.}$
$\textbf{Hence B is Answer.}$
