On the new years Eve , every member of a community exchanged cards with every other member. if a total of 420 different cards were exchanged , then how many different members were there in the community ?
but my answwer is wrong
I did it in this way
like there are five member
A B C D F
A=4
B=3
C=2
D=1
total no the cards exchanged =10
On
The key word here is exchanged. That implies that for each pair of members, say person $A$ and person $B$, there are two cards that switch hands: $A$ gives a card to $B$, and $B$ gives a card to $A$. Your counting method assumes that there is only a single card transferred between each pair of people.
On
let say there are 3 people A,B,C. Combinations possible will be 3C2 = 3 (AB,AC,BC) and there will be 6 cards exchanged i.e 3C2 x 2 ...Similarly if n people are there combinations possible will be nC2 and cards exchanged = nC2 x 2...But nC2 x 2 = 420 .. nC2 = 210...n(n-1)=420..Hence answer will be 21
If there are $n$ members, each will send $n-1$ cards. So the total cards sent is $$ n(n-1)=420 $$ $$ n^2 - n -420 = 0 $$ $$ (n-21)(n+20) = 0 $$ So $n$ is $21$ (since it obviously needs to be positive).
Note that if A and B "exchange cards," 2 different cards are exchanged, not 1. If the problem said 420 pairs of people exchanged cards, then the answer would have been 41.