Ways and then realize that it doesnt matter what order

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5x5x4x4x3x3x2x2x1x1=5!x5! ways, and then realize that it doesn’t matter what order they are put on the dance floor. Either way, we get 5 5 5 120 ! ! ! x5! = = .
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7. Any time person j shakes hands with person k , the number two will be added to the total number of handshakes. Therefore the total number of handshakes must be even. However, the total number of a handshakes = the sum of all the handshakes made by each person. Let x(j)= the number of handshakes made by person j. x j j ( ) = = total number of handshakes. 1 6 Let S = {j: x(j) shakes hands an odd number of times}. Claim: Card(S) is even. Proof: If card (S) is odd, then the number of people who shake hands an odd number of times is odd. These people will contribute an odd number of handshakes to the total number of handshakes because the sum of an odd number of odd numbers is odd ( try and prove this). The remaining people shake hands an even number of times and will contribute an even number to the total number of handshakes. Thus, the total number of handshakes = even + odd = odd number, contradicting that the total number of handshakes is even. 8. With one cut, there will be two slices. You want the second cut to intersect the first cut so you will get 4 slices. If not then you will get 3 slices. You will want the third cut not to pass through the intersection of the first two cuts in order to get the maximum of seven slices. If the third cut passes through the first two cuts , then you will get six slices. According to this scheme, the maximum number of slices is obtained when no three lines are concurrent(i.e. three lines do not intersect in a point). Note that when the third line cuts the other two lines, three new regions (pizza slices) are created if it doesn’t cross through the intersection. The fourth cut should cross the other three in a way such that no three lines are concurrent. This will create 4 new regions (pizza slices) and yield a maximum of 11 slices. Finally, the fifth cut should cross the other four cuts in a way such that no three lines are concurrent. This will create 5 new regions ( pizza slices) and yield a maximum of 16 slices.
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In general, let a(n) = maximum number of slices that can be obtained from a pizza when n cuts are made. Our analysis has shown that, a(1)=2, a(2)=a(1)+2=4, a(3)=a(2)+3=7, a(4)=a(3)+4=11, a(5)=a(4)+5=16,…, a(n)=a(n-1)+n. This recursion can be solved, and you will learn how to do this later in the course. For now, I will give you the answer, it is a(n) = 1 + n 2 + n 2 . Try and prove this by induction. 9. A n = = + + { . .
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  • Fall '06
  • miller
  • Natural number, Prime number, positive integer

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