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\centerline{HOMEWORK SET \#12 / CO1A /  Spring 2020 } \vskip3mm
\rightline{as was the FINAL / CO1A   / Spring 2019  \hskip 2cm  }

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\noindent {\it The maximum number of points given to any problem
is indicated at the end of the problems in {\bf (boldface)}. The
maximum number of points given to the whole final would be 72, but now is only 65, as problem 9 is 
listed here just for your information, but NOT TO BE SOLVED as the background material is has not been covered yet}
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\item{1.)}  Prove that $$ {(m+n)!\over m!n!} \le {(m+n)^{m+n}\over
m^m n^n}$$ Hint: use binomial theorem!
{\bf (7 points)}



\item{2.)}  Ramsey quiz: Find the following Ramsey numbers (without justification, just give the values):
$R(P_4,C_6)$, $R(C_4,C_5)$, $R(K_3,C_6)$. {\bf (3+5+4=12 points, but -1 point for each  bad answer!!!!)}



\item{3.)}  Is there any graph isomorphic to its complement on
$n$ vertices for $n=4,5,6$? {\bf (6 points)}


\item{4.)} Suppose that $a_{n+1}=(n+1)b_n$, with $a_0=b_0=1.$
Let $A(x)$ denote the exponential generating function of the
sequence $(a_n)$ and $B(x)$ denote the exponential generating
function of the sequence $(b_n)$. Derive a relation between $A(x)$
and $B(x)$. {\bf (6 points)}


\item{5.)} In how many ways can one sit $n$ couples around a round table alternating if no spouses from the same couple can
be seated next to each other.  {\bf (8 points)}


\item{6.)}  In the Hungarian TOTO (Italian Calcio) game one may bet
on the outcome of the soccer games: 1 for the win of the first
team, 2 for the win of the second team, x for a draw. How many
tickets do you have to fill if there are three matches to bet on
and you surely want to have a ticket which misses all the final
results? {\bf (10 points)}


\item{7.)}  A social worker has to make altogether 60 visits, at
least one on each day. Is there a period of consecutive days on
which he makes exactly 15 visits if he makes his visits on a) 31
days b) 30 days? {\bf (9 points)}


\item{8.)} You have $n$ forints. Every day you buy exactly one of
the following products: pretzel (1 forint), candy (2~forints),
ice-cream (2~forints), soda (2~forints), bubble gum (2~forints),
cracker (2~forints), chocolate bar (3~forints), mineral water
(3~forints) or sandwich (3~forints). What is the number of
possible ways of  spending all the money (the order of the bought
products counts)?   {\bf (7 points)}%


\item{9.)} {\bf Listed only  for the sake integrity, but not to be solved!!} \ \ \ Prove that in a $(b,v,k,r,\lambda)$ balanced incomplete block design over a set of size $v$, each block having size $k$ and
every pair of elements is in exactly $\lambda$ blocks all points are in the same number of blocks (denoted by $r$) and derive a formula for $r$ in terms of $v,k,\lambda$. {\bf (7 points)}




\bye

