Probability and Statistical Inference. Robert Bartoszynski

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      1 3.4.1 Show that if then(i) Use the definition of binomial coefficients as ratios of the factorials. (ii) Use directly the interpretation of the binomial coefficients as the number of subsets of a given size. (iii) How many ways can one choose an ‐element subset from a ‐element subset from a ‐element subset from a ‐element subset from a element set? (where ).

      2 3.4.2 Find the coefficient of the term in the expansion of .

      3 3.4.3 Use the argument analogous to that in Theorem 3.3.2 to show that if , and , then

      4 3.4.4 Use Stirling's formula to approximate the number of ways: (i) A set of size can be partitioned into two equal parts. (ii) A set of size can be partitioned into three equal parts.

      5 3.4.5 Approximate the probability that every state will be represented in the committee in Problem 3.3.1 (i).

      1 1 The integer part of , [], is the largest integer not exceeding ( etc.).

      2 2 This property, called exchangeability of events, will be discussed in more detail in Section 4.6.

      3 3 The discarded cards are not mixed with the deck. Assume that the player receives the replacement of the discarded cards from the unused remainder of the deck.