A certain group of friends met on a new year eve party and each person shook hands with everybody else in the group exactly once and the number of handshakes turned out to be 66. On the occasion of Pongal (harvest festival), if each person in this group sends a greeting card to every other person in the group, then how many cards are exchanged?
Question
A certain group of friends met on a new year eve party and each person shook hands with everybody else in the group exactly once and the number of handshakes turned out to be 66. On the occasion of Pongal (harvest festival), if each person in this group sends a greeting card to every other person in the group, then how many cards are exchanged?
Solution
The problem is about combinations. In a group of n people, the number of handshakes is given by the combination formula nC2 = n*(n-1)/2.
Step 1: We know that the number of handshakes is 66, so we can set up the equation n*(n-1)/2 = 66.
Step 2: Solving this equation gives us n^2 - n - 132 = 0.
Step 3: Factoring this equation gives us (n-12)(n+11) = 0.
Step 4: The solutions to this equation are n = 12 and n = -11. Since the number of people can't be negative, we discard n = -11. So, there are 12 people in the group.
Step 5: Now, if each person sends a greeting card to every other person in the group, the number of cards exchanged is also given by the combination formula nC2 = n*(n-1)/2.
Step 6: Substituting n = 12 into the formula gives us 12*11/2 = 66 cards.
So, 66 cards are exchanged.
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