**Unformatted text preview: **MATH 340 Discrete Stuctures 2 Winter 2019 Assignment 1
Due date: Thursday, February 7, 2019 at 4:05. Solutions You can work out the problems with your friends, but every student must submit their own
solutions, written in their own words: otherwise it may be considered as plagiarism. Late
submissions will be penalized by 10% o per day after the deadline. Paper submissions
(in class) are strongly encouraged.
Question 1 (20 marks) A website asks you to login in order to be able to write on its guestbook. But of course,
it won't let you choose just any string as a password: that would be too simple! Instead,
the requirements are the following:
1. The password must contain between 8 and 10 characters.
2. Each character must be of one of the following types: lowercase letters without
accents (az), uppercase letters without accents (AZ), digits (09) or special characters from the following list: !?#&@%*~$_
3. The password contains at least one character of at least three of the four types
mentioned above.
In how many ways can a password be chosen in order to meet all those requirements? That way of thinking about internet security is completely obsolete, but somehow
still common. XKCD has a comic strip about that... See
Note: Solution: Let's rst name and count each set of characters...
L = {Lowercase letters}
U = {Uppercase letters}
D = {Digits}
S = {Special characters} |L| = 26
|U | = 26
|D| = 10
|S| = 10 Let pk be the number of valid passwords, for k = 8, 9, 10. We will count each pk separately,
so that by the sum principle, the total number of valid passwords is p8 + p9 + p10 .
There are 2 × 26 + 2 × 10 = 72 admissible characters in total, so 72k possible strings of
length k, and from that, we want to subtract the number qk of strings that just use one or
two types of characters. So pk = 72k − qk by the complement principle.
Now, to count qk , we need to split in cases according to the two sets of characters that are
used. • |L ∪ U | = 52, so there's 52k strings made just with lowercase and uppercase letters.
• |D ∪ S| = 20, so that's 20k strings made just with digits and special characters.
• |L ∪ D| = |L ∪ S| = |U ∪ D| = |U ∪ S| = 36, so that's 36k strings made just with two of those types of characters, for each of the four possibilities. Note that the strings made with just one type of character were counted exactly 3 times
in the cases above, for each type of chracter. So we need to remove two times the (26k +
26k + 10k + 10k ) strings made of just one type of character from the total above in order
to count them just once.
Combining all of this together, we nd:
pk = 72k − qk = 72k − 52k − 20k − 4 · 36k + 4 · 26k + 4 · 10k . Substituting values for k, we nd p8 = 658270076405760, p9 = 48833762118635520 and
p10 = 3585280971297792000, for a total of 3634773003492833280 valid passwords.
Question 2 (20 marks) A wedding organizer needs to sit 4 couples and 4 single persons around a round table. In
how many ways can she sit those 12 people so that:
(a) All couples sit together (the two members of each couple sit side-by-side)?
(b) No couple sit together? Hint: Inclusion-Exclusion... Since the table is round, dierent rotations of the same arrangement are indistinguishable, but a clockwise arrangement is distinguishable from its counterclockwise version.
Note: A
D D
B = C C D
A 6= A B C
B Solution:
(a) Let us compute the number N (k) of ways for k couples to sit together, then we will
just need to set k = 4 to answer the question.
To handle the dierent rotations of the table, let us x the position of one of the
four single people, and sit everyone else starting from the right of that person (so
it's like sitting them on a row).
If k couples are to sit together, each of them can have a conguration M F or F M ,
so that's 2 options for each: 2k options. Once this is decided, we can see each couple as forming a single block, along with the
(11−2k) people that do not belong to any of those k couples. So that's k+(11−2k) =
11 − k blocks to place on a line, and there's (11 − k)! ways to do that.
Therefore, N (k) = 2k (11 − k)!. With k = 4, that's 24 (11 − 4)! = 16 × 7! = 80 640
possibilities.
(b) Let Ai be the set of congurations of the table such that the i-th couple sit together,
for i between 1 and 4. If we take the intersection of any k of those sets together,
then the cardinality is the number N (k) calculated in part (a).
The number of congurations that belong to none of those sets Ai is then given by
the inclusion-exclusion formula:
4
X
4
X
4
k 4
N (k) =
2k (11 − k)!
(−1)
(−1)
k
k
k=0
k=0
k = 11! − 4 · 2 · 10! + 6 · 4 · 9! − 4 · 8 · 8! + 16 · 7!
= 18 385 920 .
Question 3 (20 marks)
(a) Give an algebraic proof of the following identity.
2n
n
=2
+ n2
2
2
(b) Give a combinatorial proof of the same identity. Solution:
(a) Write out the denition in terms of factorials and simplify. The left hand side gives
us
2n
2 = 1
· 2n · (2n − 1) = n · (2n − 1) = 2n2 − n
2 The right hand sides gives us
1
n
2
+ n2 = 2 · · n · (n − 1) + n2 = n · (n − 1) + n2 = 2n2 − n
2
2 So the two are equal.
(b) The LHS is clearly the number of ways to choose two numbers from the set S = {1, 2, 3, . . . , 2n}. Let's see that the RHS also counts this. Divide
the set S into A = {1, 2, 3, . . . , n} and B = {n + 1, n + 2, . . . , 2n}. There are
n
ways to choose two numbers from A, n2 ways to choose two numbers from B ,
2
and n2 ways to choose one number from A and one number from B . This covers all
ways to chose two numbers from S . Question 4 (20 marks) Let (a1 , . . . an ) be any nite sequence of n integers. Show that there exist two indices s
and t, with 1 ≤ s ≤ t ≤ n, such that
t
X ai ≡ 0 mod n i=s Hint: You should use the Pigeonhole principle....
Solution: First consider the partial sums where s = 1. That is, sums of the form
1 ≤ t ≤ n.
Assume that
t
X ai ≡ rt Pt i=1 ai , where mod n i=1 If rt = 0 for any 1 ≤ t ≤ n, then we are done. So rt ∈ {1, 2, . . . , n − 1} for each
1≤tP
≤ n. Thus,Pby the pigeon-hole principle, there exist t1 < t2 such that rt1 = rt2 = r.
1
2
ai = ti=1
ai ≡ r mod n. Thus
Thus ti=1
t2
X ai − i=1 But the LHS is just Pt2 i=t1 +1 t1
X ai ≡ 0 mod n i=1 ai . So setting s = t1 and t = t2 we have
t
X ai ≡ 0 mod n i=s Question 5 (20 marks) Use the generating functions method to nd a short formula for the sum of cubes. cn = n
X k3 k=0 A(x) If A(x) is the ordinary generating function for the sequence an , then
1−x
P
is the ordinary generating function for the sequence of partial sums nk=0 ak . So if we
nd the ordinary generating function for an = n3 , we can use this trick to get the formula
for the sum of cubes.
We have
∞
∞ Solution I. X
k=0 k 3 xk = (xD)3 X
k=0 xk = (xD)3 1
1−x The expression on the right reduces as follows
(xD)3 1
x
(1 + x)
x3 + 4x2 + x
= (xD)2
=
(xD)
x
=
1−x
(1 − x)2
(1 − x)3
(1 − x)4 This is our generating function for the cubes. By the argument outlined above, if we
multiply this by 1/(1 − x), we nd the generating function for cn . So
C(x) = ∞
X cn x n k=0 We saw in class that x3 + 4x2 + x
=
(1 − x)5
∞
X
1
n+4 n
=
x .
(1 − x)5
4
n=0 Therefore,
∞
X
n+4 n
C(x) = (x + 4x + x)
x
4
n=0
!
!
!
∞
∞
∞
X
X
X
n + 4 n+3
n + 4 n+2
n + 4 n+1
=
x
+
4
x
+
x
4
4
4
n=0
n=0
n=0
!
!
!
∞
∞
∞
X
X
X
i+1 i
j+2 j
k+3 k
=
x +
4
x +
x
4
4
4
i=3
j=2
k=1
∞
X
n+1
n+2
n+3
2
= x + 9x +
+4
+
xn
4
4
4
k=3
3 2 So our expression for the sum of (for n ≥ 3) is
n
X
n+1
n+2
n+3
k =
+4
+
4
4
4
k=0
3 Which, with a little algebra becomes:
n
X k3 = k=0 n2 (n + 1)2
4 Solution II. Another way to get the closed form for C(x) is with a recurrence. We have c0 = 0 and,
for n > 0,
!
n
n−1
cn = X
k=0 k3 = X
k=0 k3 + n3 = cn−1 + n3 . Therefore,
C(x) = ∞
X cn x n = c0 + n=0 ∞
∞
X
X
(cn−1 + n3 )xn = xC(x) +
n 3 xn .
n=1 n=0 Proceeding the same way as in Solution I, we nd
∞
X n 3 xn = n=0 x3 + 4x2 + x
(1 − x)4 Therefore, when we solve the implicit equation above for C(x), we get
C(x) = x3 + 4x2 + x
(1 − x)5 The rest of the calculations go just like Solution I. ...

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- Fall '09
- KHARLMPOVICH
- Summation, Recurrence relation, Generating function, Combinatorial principles