Math 173: Final Exam practice - Spring ’09
(1) Prove (by induction) that for any integer n≥1,
1·1! + 2 ·2! + · · · +n·n!=(n+ 1)! −1.
(2) How many ways can you arrange the letters of the alphabet so that there are exactly 5
letters between the vand the t?
(3) A bag has 4 red, 4 orange, 4 green and 4 purple balls. How many distinguishable collections
of 3 balls can be chosen from this bag to be placed into a second bag?
(4) A soccer tournament has 16 teams. How many ways are there to match up the teams in 8
pairs for the first round?
(5) Using inclusion-exclusion, determine how many nonnegative integer solutions of x1+x2+
x3+x4= 12 there are in which no xiexceeds 4.
(6) Let arbe the number of solutions to 2k+p=rwhere kis a nonnegative integer and pis a
prime number. Write a generating function for {ar}and write it out through the x10 term.
What is the smallest r > 2 such that ar= 0.
(7) Use generating functions to find the number of ways to distribute rjelly beans among eight
children if:
•Each child gets at least one jelly bean
•Each child gets an even number of beans
(8) Each day you buy exactly one of the following items: tape (costing
$
1), a ruler (costing
$
1), a pen (costing
$
2), a fancy pencil (costing
$
2), paper (costing
$
2) or a loose-leaf binder
(costing
$
3). Write a recurrence relation and initial conditions for the number snof different
sequences in which you can spend exactly ndollars (n≥1).
(9) Suppose that you have an unlimited supply of red, white, blue, green, and gold poker chips
— all indistinguishable except for color. Write a recurrence relation and initial conditions
for the number cnof ways to stack nchips with no two consecutive red chips.
(10) Find a closed formula for snif s0, s1, s2, . . . is a sequence satisfying sn=sn−1+ 2sn−2for
n≥2, s0= 9 and s1= 0.
(11) Find a closed formula for snif s0, s1, s2, . . . is a sequence satisfying sn=−4sn−1−4sn−2
for n≥2, s0=−4 and s1= 2.
(12) Find a closed formula for snif s0, s1, s2, . . . is a sequence satisfying sn=sn−1−5 for n≥1
and s0= 100.
(13) Using generating functions to find a formula for snwhere s0=s1= 1 and sn=−sn−1+
6sn−2for n≥2.
(14) Give an explicit formula for snif {sn}has the generating function G(x) = 3
1−x−1
1+3x.
(15) Suppose a cake is cut into 6 identical pieces. How many inequivalent ways can we color the
cake with ncolors assuming that each piece receives one color?
(16) How many inequivalent necklaces are there which use 2 red, 2 white and 5 blue beads?
Note: the backs of the beads are identical to the fronts.
(17) Using the cycle index for the group of symmetries of the edges of the following figure (and
the theorem from Thursday’s class), compute the number of inequivalent colorings that use
at most mcolors:
(18) What is the cycle type of the permutation given in two-line notation by:
12345678
57812643