Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 668 — #676
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Chapter 15
Cardinality Rules
Rule I: Student A must not be rightmost.
Rule II: Student B must be adjacent to C (directly to the left or right of C).
Rule III: Student D is always second.
You may answer the following questions with a numerical formula that may
involve factorials.
(a) How many possible lineups are there that satisfy all three of these rules?
(b) How many possible lineups are there that satisfy at least one of these rules?
Problem 15.63.
A robot on a point in the 3-D integer lattice can move a unit distance in one positive
direction at a time. That is, from position .x; y; z/, it can move to either .x C
1; y; z/, .x; y C 1; z/ or .x; y; z C 1/. For any two points P and Q in space, let
n.P; Q/ denote the number of distinct paths the spacecraft can follow to go from
P to Q.
Let
A D .0; 10; 20/; B D .30; 50; 70/; C D .80; 90; 100/; D D .200; 300; 400/:
(a) Express n.A; B/ as a single multinomial coefficient.
Answer the following questions with arithmetic expressions involving terms n.P; Q/
for P; Q 2 fA; B; C; Dg. Do not use numbers.
(b) How many paths from A to C go through B?
(c) How many paths from B to D do not go through C ?
(d) How many paths from A to D go through neither B nor C ?
Problem 15.64.
In a standard 52-card deck (13 ranks and 4 suits), a hand is a 5-card subset of the set
of 52 cards. Express the answer to each part as a formula using factorial, binomial,
or multinomial notation.
(a) Let H be the set of all hands. What is jH j?
(b) Let H NP be the set of all hands that include no pairs; that is, no two cards in
the hand have the same rank. What is jH NP j?