On the New Years Eve, when every member exchanged cards with every other member, a total of 420 cards got exchanged; we need to find the number of members in the community.

Let the number of members in the community be ' $\(n\)$ '.


Since each member exchanged cards with every other member, the total number of cards exchanged should be $\(2\left({ }^n C_2\right)=420 \Rightarrow 2\left(\frac{n(n-1)}{2}\right)=420 \Rightarrow n(n-1)=420=21 \times 20\)$

$\(\left({ }^{\mathrm{n \mathrm{C}_{\mathrm{r=\frac{\mathrm{n}!}{\mathrm{r}!\times(\mathrm{n}-\mathrm{r})!}\right)\left({ }^{\mathrm{n \mathrm{C}_2\right).\)$ is multiplied with 2 as every members gave cards to each other, for example if there were 2 members $\(\mathrm{A} \& \mathrm{~B}, \mathrm{~A}\)$ would give card to B and B would also give card to A).

Hence the number of members in the community is 21 , so the answer is (B).