# CMPSCI 250: Introduction to Computation

### David Mix Barrington

### Spring, 2005

# Homework #5 Questions and Answers

### Question 5.2, 10 April 2005

In Problem 6.3.4, I don't understand why the strings are divided
into 111XXXXXXX, 0111XXXXXX, X0111XXXXX, ..., XXXXXX0111 instead of
111XXXXXXX, X111XXXXXX, XX111XXXXX, ..., XXXXXXX111. The latter division
makes much more sense to me.

It makes more sense to me as well, but there is a
good reason to use the less intuitive breakdown I gave you. If you use
your breakdown for eight-set inclusion/exclusion, *lots* of the
combinations of sets have nonempty intersections and the calculation gets
very complicated. With the breakdown I suggest, most of the combinations
have empty intersections and most of the terms that appear in the
inclusion/exclusion calculation are zero.

Do you see why every string with a "111" is in one of my eight sets?
Once you have counted 111XXXXXXX, you have already counted the strings in
X111XXXXXX that start with a 1, so you can restrict yourself to 0111XXXXXX.
Then once you have 111XXXXXXX and 0111XXXXXX, you already have the strings
in X1111XXXXX, so you can count X0111XXXXX instead of XX111XXXXX.

Note that a perfectly correct way to solve this problem is to write
a Java program that iterates through all 1024 binary strings of length 10,
checks each one for the presence of a 111 substring, and counts the ones that
have it.

### Question 5.1, 6 April 2005

In 6.6.4 (e), you say that a hand with three of one kind and two other
pairs (call this "AAABBCC") and a hand with two different three-of-a-kinds
(call this "AAABBBC") are to be considered full houses. What about "AAABBCD" --
is that a full house? Is it true that three of one kind, two of another, and
any two other cards constitute a full house?

There are three kinds of full houses under these rules, what you are calling
AAABBCD, AAABBCC, and AAABBBC. It is not true that anything of the form AAABBXX
is a full house, because if the X's include a fourth A or two B's, you have
AAAABBX or AAABBBB, and each of these hands is considered four of a kind rather
than a full house.

The basic rule is that you look at all the five-card subsets of a seven-card
hand, and choose the one that gives you the highest-ranking hand.

Last modified 6 April 2005