Counting Problems; and Recursion! CSCI 2824, Fall 2012!

Transcription

1 Counting Problems; and Recursion! CSCI 2824, Fall 2012!!!

2 Assignments To read this week: Sections (Ensley/Crawley Problem Set 3 has been sent out today. Challenge problem today!!!!!

3 So, to recap Order matters, repetition allowed ( password problem) Order matters, repetition not allowed ( traveling salesman problem) Order doesn t matter, repetition not allowed ( committee problem) Order doesn t matter, repetition allowed ( mixed nuts problem)

4 Practice Problems Problem 1: How many potential Mastermind codes are there (6 colors of pegs) with 2 or 3 pegs the same color? Problem 2: How many poker hands are there with one pair? Problem 3: Suppose you have 10 disks all of the same size (indistinguishable). How many ways are there of placing all these disks on the three Tower of Hanoi pegs?

5 Problem 1 No. of 2-of-a-kind No. of 3-of-a-kind (C(4,2) * 6 * 5 * 4) + (C(4,3) * 6 * 5)

6 Problem 1 (alternative) All the ways of getting 4 of a kind: 6 All the ways of getting 4 distinct: 6 * 5 * 4 * 3 All the ways of getting 2 of a kind in the form 2+2 (as opposed to 3+1): (1/2) * C(4,2) * 6 * 5 Now subtract the sum of all these from 6 4

7 Problem 2 (13 * C(4, 2)) * (C(12,3) * 4 * 4 * 4)

8 Problem 3 This is a mixed nuts problem (think of the pegs as meaning almonds, Brazils, cashews, and we have to fill a bag of 10 items): C(10+2, 2) = C(12, 2) = 66

9 Another type of problem How many numbers, from 1 to 100, are not multiples of 2, 3, or 5?

10 We could start with a simpler problem (as Polya suggests) How many numbers between 1 and 100 are neither multiples of 2 or 3? (2 or 5? 3 or 5?)

11 What we re after here is a set union The union of all multiples of 2, multiples of 3, and multiples of 5 chosen from 1 through 100. The way to do this is to (a) take the sum of the sizes of all sets, (b) subtract the sum of all 2-set intersections, then (c) add back the size of the 3-set intersection

12 Inclusion/Exclusion The general idea here is that, to get the size of the union of n sets, you need to calculate: Add C(n,1) sets (that is, all the original sets) Subtract C(n,2) 2-set intersections Add C(n,3) 3-set intersections Subtract C(n,4) 4-set intersections Add (-1) n-1 C(n,n) n-set intersections

13 Suppose an item is in the intersection of m sets. How many times will it be added in? C(m, 1) C(m, 2) + C(m, 3) +. (-1) m-1 C(m,m) Recall from our discussion of Pascal s triangle: 0 = Σ 0<=k<=m C(m,k) x k So we re adding in our item exactly 1 (that is, C(m,0) ) times, since C(m,0) is the only term missing from that top line!

14 A Practice Problem How many poker hands have at least one card of every suit?

15 Challenge Problem No. 3 (due Thursday Nov. 1) The Secret Santa problem: Suppose we have 8 children in a classroom, and we want to assign each of them a Secret Santa. Naturally, we don t want to assign any kid to be his or her own Santa. How many distinct legal arrangements are there? And now that you ve solved the problem for 8, write an expression that solves the problem for n in general. As n gets large, what is the probability that any Secret Santa arrangement will be a legal one?

16 RECURSION: the basic definition In a computer program, a procedure or function is recursive if its definition involves a call to itself. There are slightly more broad definitions that still make sense: for instance, a set of functions or procedures that mutually call each other can be said to be recursive as well. Or we might say of a definition of some sort of object that the definition itself has some element of recursion.

17 A 0-level cat is a cat wearing a hat. A n-level cat is a cat tipping its hat to reveal an n-1 level cat.

18 A well-known recursive definition in mathematics Factorial (n) IF (n = 0) THEN 1 ELSE ( n * Factorial(n -1) )

19 Another recursive definition: here, assume n1 and n2 are nonnegative integers Add (n1 n2) IF (= n2 0) THEN n1 ELSE Add (Increment n1) (Decrement n2)

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24 Our earlier recursive solution to the Tower of Hanoi puzzle: To solve the puzzle with 1 disc, just move the disc directly. To solve the n-disc puzzle with pegs labelled SOURCE, SPARE, TARGET: Solve the (n-1) disc puzzle with pegs relabelled SOURCE TARGET SPARE Move the largest (bottom) disc from SOURCE to TARGET Solve the (n-1) disc puzzle with pegs relabelled SPARE SOURCE TARGET

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