HMC Math 55: Discrete Mathematics (Fall, 1998)
Homework

### Assignment #4 (due Wed. 9/30):

Section 15 (pg 103) (Prove these combinatorially): 4 (Optional hint: *'s & |'s), 6
Section 16 (pg 112): 1, 4
4A: For each of the following "stars & bars" notations, name the corresponding multiset and the set from which it was chosen (i.e., for "*|**|", the correct answer is "<1,2,2> from {1,2,3}").
(i) | | | |*****
(ii) ***| | |**
(iii) | |*****| | |
(iv) *| |**| |**| |
(v) *|**|***|
(vi) *****
4B: Find some function f : Z->Z+ (from the integers to the positive integers) which is a 1-1 correspondence. Prove this (i.e. prove f is 1-1 and onto).
4C: In how many ways can n identical chemistry books, r identical math books, s identical physics books and t identical biology books be arranged on 3 (distinct) bookshelves? (Any shelf may get any number of books)

### Assignment #3 (due Wed. 9/23):

Section 14 (pg 95): 13
Section 18 (pg 130): 3, 4 (prove these by induction, not the techniques of Section 18)
Section 19 (pg 138): 4b, 4e

3E: How many poker hands are superior to 3 Aces? That is, how many hands are straights, flushes, full houses, 4-of-a-kinds or straight flushes? Assume you're being delt a single hand from a full 52 card deck, and don't forget to substract off the straight flushes! (Why?)

Extra Credit: Section 14, #18.
Your output may be in simple ASCII, and may use *'s or something similar. An equispaced font is highly recommended.

### Assignment #2 (due Wed. 9/16):

Section 6 (pg 37): 7 ("How many possible combinations are there?"), 10
Section 10 (pg 63): 3, 8, 13
Section 14 (pg 95): 2, 9, 12, 15

### Assignment #1 (due Fri. 9/11):

Section 2 (pg 13): 4, 8
Section 3 (pg 20): 8
Section 4 (pg 23): 1, 5
Section 5 (pg 27): 8, 11b
Section 6 (pg 37): 2, 6
Section 9 (pg 52): 1g, 1j, 2g, 2j

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