1
Mat 344F challenge set #2
Solutions
1.
Put two balls into box 1, one ball into box 2 and three balls into box 3.
The remaining 4 balls can now be distributed in any way among the
three remaining boxes. This can be done in C(4+3-1,4)=C(6,4)=15
ways. Please note that the question should have said that box 1 has at
least two balls.
2.
There are mn entries in the m x n matrix, two choices for each, and the
choices are independent of one another, hence by the multiplication
principle, there are 2
mn
matrices that have 0,1 entries. If we allow the
entries to be 0,1,2,3 then each of the mn entries has four choices which
are independent of one another. The multiplication principle tells us that
there are 4
mn
such matrices.
3.
Let A be the event,” exactly one pair of shoes is selected”. Let B be the
event “exactly two pairs of shoes are selected. Let S be the event “ four
shoes are selected from 20 shoes with order being irrelevant”. We want
to calculate P(at least one pair of shoes is selected) = P(A or B) = P(A)
+P(B) since events A and B are disjoint. We will use the probability
formula
P(A) =
n(A)
n(B)
.
The number of ways to select exactly one pair of
shoes = n(A) = C(10,1) x (C(18,2)- 9). C(10,1) counts the number of
ways to pick one pair of shoes. There are 18 shoes remaining, pick two
shoes. However we don’t want to pick another pair so we must subtract
the 9 remaining pairs of shoes. The number of ways of picking two
pairs of shoes = n(B) = C(10,2). Using these results, it follows that
P(selecting at least one pair of shoes) =
C
C
(
, )
(
, )
.
10 1
20 4
x(C(18,2) - 9)
C(10,2)
C(20,4)
+
Another way to approach this problem is to first calculate the
probability that no pair of shoes is selected and then subtract this from
one. This will yield the probability that at least one pair of shoes is
selected. The number of ways to select no pair of shoes is C(10,4) x
C(2,1) x C(2,1) x C(2,1) xC(2,1). C(10,4) counts the number of ways to
select four shoes out of ten with order being irrelevant. Once the pairs
have been selected, then out of each pair we can select one shoe, and this