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## PowerPoint Slideshow about 'Handshake Problem and Phone Call Problem' - axel

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Handshake Problem### Handshake Problem and Phone Call 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?

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?

+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

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

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