How Many Handshakes?Date: 08/18/2005 at 20:39:33 From: Victor Subject: How Many Handshakes Thirty people at a party shook hands with each other. How many handshakes were there altogether? Before answering this question, draw a diagram and see if you can establish a pattern by collecting data in a table for one, two, three, four and five people shaking hands. If there were 300 people at the party, how many handshakes will there be altogether? I'm not sure how to draw a diagram establishing a pattern of the number of people and the number of handshakes. Date: 08/19/2005 at 16:01:51 From: Doctor Wilko Subject: Re: How Many Handshakes Hi Victor, Thanks for writing to Dr. Math! First, when we say handshakes, we'll agree that we mean Adam shaking Bob's hand is the same as Bob shaking Adam's hand, i.e., once two people shake hands it is considered a hand shake. Now let's reason through the handshakes, starting with two people and working our way up, while looking for a pattern. If there are two people at a party, they can shake hands once. There is no one else left to shake hands with, so there is only one handshake total. 2 people, 1 handshake If there are three people at a party, the first person can shake hands with the two other people (two handshakes). Person two has already shaken hands with person one, but he can still shake hands with person three (one handshake). Person three has shaken hands with both of them, so the handshakes are finished. 2 + 1 = 3. 3 people, 3 handshakes If there are four people at a party, person one can shake hands with three people, person two can shake hands with two new people, and person three can shake hands with one person. 3 + 2 + 1 = 6. 4 people, 6 handshakes Are you seeing a pattern? If you have five people, person five shakes four other hands, person four shakes three other hands, person three shakes two other hands, and person two shakes one hand. Another way to see it is, Person 5 Person 4 Person 3 Person 2 4 + 3 + 2 + 1 = 10 handshakes total ==================================================== People at Party Number of Handshakes 2 1 3 1 + 2 = 3 4 1 + 2 + 3 = 6 5 1 + 2 + 3 + 4 = 10 6 1 + 2 + 3 + 4 + 5 = 15 . . . n(n-1) n 1 + 2 + ...+ (n-1) = -------- 2 ===================================================== It turns out this is just the formula for adding up an arithmetic sequence where you know how many terms you have total, and you know the first and last terms of the sequence. See this link for more on that: Adding Arithmetic Sequences http://mathforum.org/library/drmath/view/55724.html So, to find how many handshakes there are at a party of 30 people, you could add up all the numbers from 1 to 29 or use the formula, (30)(29) ---------- = 435 handshakes 2 Can you figure out how many handshakes there are at a party of 300 people? Here's another link from our archives: Handshakes at a Party http://mathforum.org/library/drmath/view/56139.html Does this help? Please write back if you have any further questions. - Doctor Wilko, The Math Forum http://mathforum.org/dr.math/ Date: 08/19/2005 at 23:49:46 From: Victor Subject: Thank you (How Many Handshakes) Thank you very much Dr. Wilko! |
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