CSE 21: Quiz 2 Solutions
August 14, 2007
1. How many ways to deal 5 out of 52 cards such that you have two pairs? Note
that a full house (3 of one rank, 2 of another) doesnโt count as two pair.
Answer: We first choose the two denominations for our 2 pair ๎13
2๎, and
then the 2 suits for each of our pairs ๎4
2๎2, and finally we must choose 1
of the 44 remaining cards that are not of our 2 denominations as the 5th
card in our hand ๎44
1๎. Putting it all together gives ๎13
2๎๎4
2๎2๎44
1๎.
2. A bakery produces seven different kinds of pastry (one of which is an eclair).
If the different kinds of selections of 15 pastries are equally likely, what
is the probability that a selection contains exactly 4 eclairs? This is a
stars and bars problem. The total number of ways to select 15 pastries
unconstrained is equivalent to having 15 stars (pastries) and 7 (kinds of
pastry)-1=6 bars giving ๎21
6๎. To have exactly 4 eclairs is the same as
throwing 4 eclairs into our selection and then removing eclairs as a choice
for the remaining 11 pastries. This gives 11 stars and 6 - 1 = 5 bars
and ๎16
5๎ways to have exactly 4 eclairs. Putting the two together gives a
probability of (16
5)
(21
6)
1