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## Handshake Problem and Phone Call Problem

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**Handshake Problem**If there are 30 people in a room and everyone has to shake hands, how many handshakes will there be? What about if there are n people in the room? Handshake Problem and Phone Call Problem Phone Call Problem 30 people are invited to a party. If every person speaks to every other person on the phone beforehand, how many phone calls will there be? What about if there are n people invited to the party?**Student 1 ViewPoint – Start Simple!**0 1 3 6 10 4 people 1 person 2 people 5 people 3 people**+1**+2 +3 +4 +5 +1 +1 +1 +1 Sequence: 0, 1, 3, 6, 10, 15, ... Since the second difference is constant, we have a quadratic sequence with first term ½ n² ½ n² Compare Original Sequence with Sequence: 0, 1, 3, 6, 10, 15, ... ½ n²: 0.5 2 4.5 8 12.5 18 -0.5, -1, -1.5, -2, -2.5, -3, ... T(n) = -½n Therefore T(n) = ½ n² - ½n**Questions you might ask yourself?**• What do T(n) and n represent? 2) Does the formula work?**Student 2 Viewpoint**Start with less people. For example when n = 6 If there are 6 people then every single person will have to make 5 phone calls. 6 x 5 = 30 calls However, this is twice as many calls as is needed because if you´ve already been called by someone then you don´t need to call them back. Therefore, the number of calls is: 6 x 5 = 15 calls 2**Student 2 Viewpoint**Hence for n = 7 people, the number of calls is: 7 x 6 = 21 calls 2 For n = 8 people, 8 x 7 = 28 calls 2 For n people, n x (n-1) = Number of calls 2**Student 2 Viewpoint (without words)**6 x 5 = 30 calls 6 x 5 = 15 calls 2 7 x 6 = 21 calls 2 8 x 7 = 28 calls 2 n x (n-1) = Number of calls 2 Which is easier to understand?**Student 3 viewpoint**Number of Telephone Calls = What does this mean? Out of n objects, how many ways are there to choose 2 of them? E.g. If you have one object: can´t choose two of them! two objects: 1 way to choose three objects: 3 ways to choose four objects: 6 ways to choose five objects: 10 ways to choose etc. Lisa & Bart Lisa & Homer Bart & Homer Lisa & Marge Bart & Marge Marge & Homer Lisa & Maggie Bart & Maggie Marge & Maggie Homer & Maggie So we do have the triangle number sequence again: 1, 3, 6, 10, 15, ...**Why does = n(n-1) ? 2**= n! 2!(n-2)! = n x ( n-1) x (n - 2)! 2! (n – 2)! = n(n – 1) 2 11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6 15 20 15 6 11 7 21 35 35 21 7 11 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 1