14.11. References 599
Problem 14.21.
Answer the following questions with a number or a simple formula involving fac-
torials and binomial coefficients. Briefly explain your answers.
(a)How many ways are there to order the 26 letters of the alphabet so that no two
of the vowelsa,e,i,o,uappear consecutively and the last letter in the ordering
is not a vowel?
Hint:Every vowel appears to the left of a consonant.
(b)How many ways are there to order the 26 letters of the alphabet so that there
areat least twoconsonants immediately following each vowel?
(c)In how many different ways can2nstudents be paired up?(d)Twon-digit sequences of digits 0,1,... ,9 are said to be of thesame typeif the
digits of one are a permutation of the digits of the other. FornD 8 , for example,
the sequences 03088929 and 00238899 are the same type. How many types of
n-digit sequences are there?
Problem 14.22.
In a standard 52-card deck, each card has one of thirteenranks in the set,R, and
one of foursuits in the set,S, where
RWWDfA;2;:::;10;J;Q;Kg;
SWWDf|;};~;g:A 5-cardhandis a set of five distinct cards from the deck.
For each part describe a bijection between a set that can easily be counted using
the Product and Sum Rules of Ch. 14.1, and the set of hands matching the specifi-
cation.Give bijections, not numerical answers.
For instance, consider the set of 5-card hands containing all 4 suits. Each such
hand must have 2 cards of one suit. We can describe a bijection between such hands
and the setSR 2 R^3 whereR 2 is the set of two-element subsets ofR. Namely,
an element
.s;fr 1 ;r 2 g;.r 3 ;r 4 ;r 5 // 2 SR 2 R^3
indicates
- the repeated suit,s 2 S,
- the set,fr 1 ;r 2 g2R 2 , of ranks of the cards of suit,s, and