50 SETS Chapter 1
- (a) How many 5-card poker hands can be dealt from a deck of 52 cards?
*(b) If five cards are dealt consecutively from a standard deck, how many ways are
there to deal first an ace, second a seven, third an eight, fourth a ten, and fifth
a face card?
(c) How many 13-card bridge hands can be dealt from a deck of 52 cards?
- (a) How many sets of ten answers to ten questions on a true-false test are
possible?
(b) How many ways are there to respond to eight questions on a multiple-choice
test if each question has four choices?
(c) How many ways are there to match the first ten letters of the alphabet with
the integers 1 through 10 on a column-matching test?
- (a) How many pairs of names consisting of a female name followed by a male
name can be formed from a list of eight female and five male names?
(b) How many automobile license plates can your state issue, using its current
format for such plates?
- Suppose U = (1,2,3,... ,9, 10). How many particular cases are encompassed
by the theorems:
(a) A n (B u C) = (A n B) u (A n C) for all subsets A, B, and C of U?
*(b) A = (A n B) u (A n B') for all subsets A, B of U?
- Numbers of the form C(n, k) = n!/k!(n - k)! (0 5 k 5 n) are of considerable im-
portance. In particular, the alternative notation a, called the binomial coefficient of
n over k (or else simply n choose k), is used to denote this quantity (the name
comes from the binomial theorem whose statement involves binomial coefficients).
(a) Let n = 10 and k = 0, 3, 4, 6, 7, 10, and verify for these cases the formulas:
(b) Formula (i) of part (a) suggests that, for example, the number of three-element
subsets of an eight-element set equals the number of five-element subsets of that
set. Describe a general way of constructing a "one-to-one match-up" between
the k-element subsets of an n-element set and the (n - k)-element subsets of that
set?
- (a) Find a formula relating 2" to the binomial coefficients of the form 0, k =
0,1,2,... , n, where n E N. [Hint: Keep in mind that 2" is the total number of
subsets of an n-element set, whereas (I;) gives the number of k-element subsets
of that set, for each k = 0, 1,2,... , n.]
(b) Verify your formula from part (a) of this exercise for the case n = 10.
