If 8 people hand shake with each other, the total number of handshakes which will take place is same as the number of ways to select 2 people from $\(8={ }^8 \mathrm{C}_2=\frac{8!}{2!\times(8-2)!}=\frac{8 \times 7}{2 \times 1}=28\)$

$$
\(\left(\because{ }^n C_r=\frac{n!}{r!\times(n-r)!}\right)\)
$$


Next as people exchange phone numbers i.e. each person takes the number of each other person and gives his number to all, so the number of mobile exchange done is twice the number of ways 2 persons can be selected from 8 person i.e. $\(2\left({ }^8 \mathrm{C}_2\right)=2 \times 28=56\)$

As the same case is when cards are exchanged among 8 person, so the number of cards exchanged $\(=2\left({ }^8 \mathrm{C}_2\right)=2 \times 28=56\)$
Note: - Option (B) is wrong as 56 mobile numbers will be exchanged not 56 times.
Hence options (A) \& (D) are correct.