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\centerline{Quiz 3 Math 302}

Work any two of the three problems below.

\item{1.} A deck of cards contains 13 of each of four suits, and two jokers. The jokers can substitute for any other card in the deck. What is the probability of drawing a flush (five cards all of the same suit) in five cards from this deck? Express your answer in a form in which all that remains to be done is some multiplication and division on a calculator.

\item{2.} All parts of this question have to do with the binomial theorem.
\itemitem{(a)} State the binomial theorem, if possible in the formal language of quantifiers, sigma notation and so on. A full statement in careful (legal-style) English will also do.
\itemitem{(b)} $$\hbox{Evaluate\ }\sum_{k=0}^{10}\pmatrix{21\cr k\cr}.$$
\itemitem{(c)} $$\hbox{Evaluate\ }\sum_{k=0}^{10}\pmatrix{21\cr 2k\cr}.$$

\item{3.} How many of the binomial coefficients $\pmatrix{100\cr k\cr}$ are odd? Explain. (A full proof is not required. This task calls for reasoned speculation, not iron-clad proof.)\bye