14.11. References 619
Problem 14.57.
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.xC
1;y;z/,.x;yC1;z/, or.x;y;zC1/. For any two points,PandQ, in space, let
n.P;Q/denote the number of distinct paths the spacecraft can follow to go from
PtoQ.
Let
AD.0;10;20/;BD.30;50;70/;CD.80;90;100/;DD.200;300;400/:(a)Expressn.A;B/as asingle multinomial coefficient.
Answer the following questions with arithmetic expressions involving termsn.P;Q/
forP;Q2fA;B;C;Dg. Do not use numbers.
(b)How many paths fromAtoCgo throughB?(c)How many paths fromBtoDdonotgo throughC?(d)How many paths fromAtoDgo throughneitherBnorC?Problem 14.58.
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)LetHbe the set of all hands.What isjHj?
(b)LetHNPbe the set of all hands that include no pairs; that is, no two cards in
the hand have the same rank.
What isjHNPj?
(c)LetHSbe the set of all hands that are straights, i.e. the ranks of the five cards
are consecutive. The order of the ranks is.A;2;3;4;5;6;7;8;9;10;J;Q;k;A/;
note thatAis appears twice.
What isjHSj?
(d)LetHF be the set of all hands that are flushes; that is, the suits of the five
cards are identical.
What isjHFj?