Math 5010-1, Spring 2005
Assignment 1
Problems
#3, p. 15. Deal a job, one to each worker. This results in a total number of 20! possible job
arrangements.
#6, p. 16. To find a kitten you need to perform the following experiments:
- Choose a wife (7 ways);
- Choose a sack (7 ways);
- Choose a cat (7 ways);
- Choose a kitten (7 ways).
This makes for a total of 74 = 2401 kittens.
#21, p. 17. Call any path that goes up or right, starting from A and ending in B, an AB path.
Note that all AB paths have exactly 7 edges (and 8 vertices). To obtain all AB paths,
simply assign four →’s and three ↑’s to all vertices. [Any order will get you from A
to B, and this is the only way to do so.] Therefore, the number of AB paths is the
number of ways to choose where to place the →’s; i.e., 74 = 35.
#22, p. 17. Call the circled vertex “vertex C.” To form an ACB path you must: (i) Form an
AC path; and then (ii) continue it along a CB path. The number of ways to do (i)
is 42 = 6. The number of ways to perform (ii) is 32 = 3. Therefore, the number of
ACB paths is 6 × 3 = 18.
Theoretical Exercises
#2, p. 18. Pretend that you are going to write down all outcomes. What would you do? One
way to do that is to clump all outcomes where experiment number one is 1 together,
2’s together, etc. In the first clump
Pm we have n1 outcomes. The second clump has n2 ,
etc. All together, you will have i=1 ni = n1 + · · · + nm possible outcomes.
#3, p. 18. Once again, pretend that you are going to write down all outcomes.
Here is one way
n
to do it: First write all unordered outcomes. There are r -many of them. Each
unordered outcome can be used to create r!-many ordered ones. This leads to
n
n!
r! =
= n(n − 1) × · · · × (n − r + 1).
r
(n − r)!
Important Exercise: Come up with a proof of this formula that involves a count of
how many ways we can distribute r places among n.
#8, p. 20. Consider a group of n men and m women. The number of ways to form a team of r
people from this group is
n+m
N =
.
(EQ1)
r
1
Suppose we were to list out all such teams.
One way to do this is to clump together
n m
all ways to create a team of r women [ 0 r ways]; clump together all ways to create a
m
team of (r − 1) women and 1 man [ n1 r−1
]; etc. This leads us to
r X
n
m
N =
.
i
r
−
i
i=0
Because (EQ1) and (EQ2) must agree we are done.
2
(EQ2)