Handshake Problem and Phone Call Problem Handshake Problem

Handshake Problem and Phone Call Problem Handshake Problem a) If there are 30 people in a room and everyone has to shake hands, how many handshakes will there be? b) What about if there are n people in the room? Phone Call Problem a) 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? b) What about if there are n people invited to the party?

Student 1 View. Point – Start Simple! Number of People Number of Phone Calls 1 person 1 0 2 3 1 3 4 people 2 people 3 people 5 people 4 5 6 10

Sequence: 0, 1, 3, 6, 10, 15, . . . +1 +2 +1 +3 +1 +4 +1 +5 +1 Since the second difference is constant, we have a quadratic sequence with first term ½ n² Compare Original Sequence with ½ n² Sequence: 0, 1, 3, 6, 10, 15, . . . ½ n²: 0. 5 2 4. 5 8 12. 5 18 -0. 5, -1. 5, -2. 5, -3, . . . Therefore T(n) = ½ n² - ½n T(n) = -½n

Questions you might ask yourself? 1) 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. Lisa & Bart Lisa & Homer Lisa & Marge Lisa & Maggie Bart & Homer Bart & Marge Bart & Maggie 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. Marge & Homer Marge & Maggie Homer & Maggie So we do have the triangle number sequence again: 1, 3, 6, 10, 15, . . .

Why does = n! 2!(n-2)! = n x ( n-1) x (n - 2)! 2! (n – 2)! = n(n – 1) 2 = n(n-1) ? 2 1 11 121 1331 14641 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 84 36 9 1