Probability. Robert P. Dobrow

      and read as “ choose .” On calculators, the option may be shown as “.”

TABLE 1.4. Common values of binomial coefficients.

Binomial coefficients

The binomial coefficient is often referred to as a way to count combinations, numbers of arrangements where order does not matter, as opposed to permutations, where order does matter. From the equation, one can see that the number of combinations of obtaining objects from is equal to the number of permutations of objects out of objects divided by the factorial of , the size of the subset desired (the number of permutations of the objects). This is accounting for the fact that in combinations, we do not care about the order of the objects in the subset.

      By the one-to-one correspondence between -element subsets and binary lists with exactly ones, we have the following.

      COUNTING k-ELEMENT SUBSETS AND LISTS WITH k ONES

      There are -element subsets of .

      There are binary lists of length with exactly ones.

      There are ways to select a subset of objects from a set of objects when the order the objects are selected in does not matter.

Binomial coefficients are defined for nonnegative integers and , where . For or , set Common values of binomial coefficients are given in Table 1.4.

       Example 1.20 A classroom of ten students has six females and four males. (i) What are the number of ways to pick five students for a project? (ii) How many ways can we pick a group of two females and three males?

       Example 1.21 In a poker game, players are dealt five cards from a standard deck of 52 cards as their starting hand. The best hand in the game of poker is a royal straight flush consisting of 10-Jack-Queen-King-Ace, all of the same suit. What is the probability of getting dealt a royal straight flush?There are four possible royal straight flushes, one for

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