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Counting Problems
CTI, Nov. 6, 2015
Let’s play the following game.1 I’ll write a positive integer on the board
that comes from a set S. You can propose other numbers, and I tell you if
your proposed number comes from the set. Eventually you will figure out
the properties that define the set. Later in our work, we’ll count the number
of members of the set. We will also repeatedly phrase each problem in the
language of finite probability just to emphasize the similarities. 2
1. All three-digit numbers.
2. Four digit numbers that include the digit 5. The wording of the analogous probability problem would be: what is the probability that a
randomly selected four digit number has the digit 5 among its digits?
3. Three digit multiples of 7.
4. Five digit numbers with exactly two 3’s and three 5’s.
5. Five digit numbers which don’t include the digit 0 and have five different digits.
6. Five digit numbers that peak in the middle.
7. Five digit numbers that use only the digits 3 and 5.
8. Five digit numbers that have no repeated digits.
9. Four digit numbers that use just the digits 6, 7, and 8.
10. Seven digit numbers consisting only of the digits 6, 7, and 8 and satisfying the property that any two consecutive digits differ by 1.
11. Numbers (any length) that use all different digits.
12. Three digit numbers for which the sum of the digits is 9.
13. Four digit numbers that use four different digits and can begin with 0
(thus including three-digit numbers that do not include the digit 0.
14. Five digit palindromes.
1
2
c
Unauthorized reproduction/photocopying prohibited by law’ ⃝
c
Unauthorized reproduction/photocopying prohibited by law’ ⃝
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Counting Problems
CTI, Nov. 6, 2015
15. Four digit numbers for which the digits increase from left to right,
sometimes called rising numbers.
16. Four digit numbers for which the digits decrease from left to right,
sometimes called falling numbers.
17. Four digit numbers for which the numbers do not decrease, sometimes
called non-decreasing numbers.
18. Six digit numbers a1 a2 a3 a4 a5 a6 that use just the digits 1 through 6, each
just once and satisfy the requirement a1 < a2 > a3 < a4 > a5 < a6 .
These are called up-down numbers.
We’re going to start with some easier problems. For convenience we are
discussing numbers that can be built without the digit 0. In other words, we
can use only digits from D = {1, 2, 3, 4, 5, 6, 7, 8, 9}. These items are counted
in the chart below.
1. How many four-digit number are there?
2. How many four-digit numbers have four different digits?
3. How many four-element subsets does the nine-element set
D = {1, 2, 3, 4, 5, 6, 7, 8, 9} have? Alternatively, how many four-digit
increasing numbers are there?
4. How many how many four-digit non-decreasing numbers are there?
( )
n!
Throughout we use both the notations nr and Crn for the number (n−r)!r!
.
We are ready to discuss the general idea of counting samples taken from a
population of objects. In doing such sampling we are allowed to make a
distinction between the order in which the objects became a part of the
sample or not. We are also allowed to sample with replacement or not.
This leads to four different types of samples. If we count as different two
samples that have the same elements but in a different order, we call these
arrangements, and if we don’t distinguish on this basis, we call the samples
selections. Let’s classify each of the counting problems above using these two
questions.
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CTI, Nov. 6, 2015
order matters
order does not matter
conditions
arrangements()
selections{}
with repetitions
Exponations
Yahtzee Rolls
(
)
Ern = nr
Y(rn = C)rn+r−1 = n+r−1
r
all 94 = 6561
all 9+4−1
= 495 four-digit
4
four-digit numbers
nondecreasing numbers
without
Permutations
Combinations( )
n!
n!
n
n
repetitions
Pr = (n−r)!
Cr = (n−r)!r!
= nr
()
all P49 = 9!/5! = 3024 four-digit
all 94 = 126 four-digit
numbers with four different digits
increasing numbers
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Counting Problems
CTI, Nov. 6, 2015
1. (1998 state math contest) There are 8 girls and 6 boys in the Math
Club at Central High School. The Club needs to form a delegation to
send to a conference, and the delegation must contain exactly two girls
and two boys. How many delegations that can be formed?
2. (1998 state math contest) Given n a positive integer, a plus or minus
sign is assigned randomly to each of the integers 1, 2, . . . , n. Let P (n)
be the probability that the sum of the n signed numbers is positive.
What is the value of [P (1) + P (2) + · · · + P (6)]?
3. (1999 state math contest) A box contains b red, 2b white and 3b blue
balls, where b is a positive integer. Three balls are selected at random
and without replacement from the box. Let p(b) denote the probability
that no two of the selected balls have the same color. Is there a value
of b for which p(b) = 1/6?
4. (2009 Mathcounts) Four points A, B, C and D on one line segment are
jointed by line segments to each of five points E, F, G, H, and I on a
second line segment. What is the maximum number of points interior
to the angle belonging to two of these twenty segments.
I..
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5. (Mathcounts 2009) How many three-digit numbers can be built from
the digits in the list 2, 3, 5, 5, 5, 6, 6?
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CTI, Nov. 6, 2015
6. A falling number is an integer whose decimal representation has the
property that each digit except the units digit is larger than the one to
its right. For example, 96520 is a falling number but 89642 is not. How
many five-digit falling numbers are there? How many n-digit falling
numbers are there, for n = 1, 2, 3, 4, 5, 6, 7, 8, and 9? What is the total
number of falling numbers of all sizes?
()
7. Cyprian writes down the middle number in each of the 95 = 126 fiveelement subsets of S = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Then he adds all these
numbers together. What sum does he get?
8. Counting sums of subset members.
(a) How many numbers can be expressed as a sum of two or more
distinct members of the set {1, 2, 3, 4, 5, 6, 7, 8, 9}?
(b) How many integers can be expressed as a sum of two or more
different members of the set {0, 1, 2, 4, 8, 16, 32}?
(c) How many numbers can be expressed as a sum of four distinct
members of the set {17, 21, 25, 29, 33, 37, 41}?
(d) How many numbers can be expressed as a sum of two or more
distinct members of the set {17, 21, 25, 29, 33, 37, 41}?
(e) How many integers can be expressed as a sum of two or more
distinct elements of the set {1, −3, 9, −27, 81, −243}?
9. How many of the first 242 positive integers are expressible as a sum of
three or fewer members of the set {30 , 31 , 32 , 33 , 34 } if we are allowed
to use the same power more than once. For example, 5 = 3 + 1 +
1 can be represented, but 8 cannot. Hint: think about the ternary
representations.
10. John has 2 pennies, 3 nickels, 2 dimes, 3 quarters, and 8 dollars. For
how many different amounts can John make an exact purchase (with
no change required)?
11. How many positive integers less than 1000 have an odd number of
positive integer divisors?
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12. (2004 AMC 10 and extensions) An 8 × 10 grid of squares with one
shaded square is given.
........................................................................
......................................................
(a) How many different squares are bounded by the gridlines?
(b) How many different rectangles are bounded by the gridlines?
(c) (∗)How many different squares bounded by the gridlines contain
the shaded square?
(d) How many different rectangles bounded by the gridlines contain
the shaded square?
13. How many squares in the plane have two or more vertices in the set
S = (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0).
14. Numbers with a given digit sum.
(a) How many numbers in the set {100, 101, 102, . . . , 999} have a sum
of digits equal to 9?
(b) How many four digit numbers have a sum of digits 9?
(c) How many integers less than one million have a sum of digits equal
to 9?
15. (2008 State Math Contest) Two coins are removed randomly and without replacement from a box containing 3 nickels, 2 dimes, and 1 coin
of value 0. What is the probability that one of the removed coins is
worth five cents more than the other?
16. (2008 State Math Contest) A palindrome on the alphabet {H, T } is
a sequence of h’s and T ’s which reads the same from left to right as
from right to left. Thus HT H, HT T H, HT HT H and HT HHT H are
palindromes of lengths 3,4,5, and 6 respectively. Let P (n) denote the
number of palindromes of length n over {H, T }. For how many values
of n is 1000 < P (n) < 10000?
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Counting Problems
CTI, Nov. 6, 2015
17. An urn contains marbles of four colors: red, white, blue, and green.
When four marbles are drawn without replacement, the following events
are equally likely:
(a) the selection of four red marbles;
(b) the selection of one white and three red marbles;
(c) the selection of one white, one blue, and two red marbles; and
(d) the selection of one marble of each color.
What is the smallest number of marbles that the urn could contain?
18. Look at the m × n multiplication table below. What is the sum of the
mn entries in the table?
× 1 2
3
4 5 6 7 8 9 ... n
1 1 2
3
n
2 2 4
6
2n
3 3 6
9
3n
..
.
m
m 2m 3m
mn
19. Consider the m × n grid of squares shown below.
How many rectangles are bounded by the gridlines?
20. How many circles in the plane contain at least three of the points
(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)?
21. The problem below appeared at the KenKen website during the week
of October 7 to 11, 2013. As counting problems often do, this one provoked some discussion about the solution.The purpose of this essay is
to solve this and some related problems.
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CTI, Nov. 6, 2015
C•
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•
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♡
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A
B
The figure above is built with nine line segments, each with either A
or B as an endpoint. Triangles and quadrilaterals of various sizes are
formed with parts of these segments as boundary.
(a) How many triangles are there?
(b) How many convex quadrilaterals are there?
(c) How many regions (triangles and quadrilaterals) have a ♡?
(d) Generalizing, suppose there are n − 2 points inserted on the segment AC and m − 2 points inserted on the segment BC. How
many triangular regions are there.
22. Consider the 9 × 9 grid of lattice points shown below. Points P and Q
are given. How many points R in the grid are there for which triangle
P QR is isosceles?
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Counting Problems
CTI, Nov. 6, 2015
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23. Next, let P = (a, b) be a lattice point. Find necessary and sufficient
conditions on P so that the set of points Q for which triangle P QO has
integer area, where O is the origin, is finite.
24. How many subsets of S = {1, 2, 3, 4, 5, 6, 7, 8} have either 5, 6 or 7 as
their largest element.
25. How many rectangular regions are bounded by the gridlines of the figure
below.
26. (Block Walking) How paths of length 15 along the gridlines start at S
and end at F the figure below. How many paths of length 17 from S
to F are there?
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Counting Problems
CTI, Nov. 6, 2015
F
S
27. (Mathcounts 2010, Target Round) Seven identical red cards and three
identical black cards are laid down in a row on a table. How many
distinguishable arrangements are possible if no two black cards are
allowed to be adjacent to each other?
28. (old USAMO problem) In a math contest, three problems, A, B, and C
were posed. Among the participants there were 25 who solved at least
one problem. Of all the participants who did not solve problem A, the
number who solved problem B was twice the number who solved C. The
number who solved only problem A was one more than the number who
solved A and at least one other problem. Of all participants who solved
just one problem, half did not solve problem A. How many solved only
problem B?
29. In a chess tournament, the number of boy participants is double the
number of girl participants. Every two participants play exactly one
game against each other. At the end of the tournament, no games were
drawn. The ratio between the number of wins by the girls and the
number of wins by the boys is 7:5. How many boys were there in the
tournament?
30. Find the number of positive integer triples (x, y, z) satisfying xy 2 z 3 =
1, 000, 000.
31. For how many different subsets of the set S = {2, 3, 4, 6, 15, 20, 30} is
the sum of the elements at least 50?
32. For each of the four given sequences try to find a pattern that is true for
all the terms of the sequence. Some sequences may have more than one
solution or there may be more than one way to describe the pattern.
(a) 1, 1, 2, 3, 5, 8, 13, 21, . . .
(b) 2.5, 4, 7, 13, 25, 49, 97, 193, . . .
(c) 1, 5, 14, 30, 55, 91, 140, 204, . . .
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CTI, Nov. 6, 2015
(d) 5, 13, 37, 109, 325, 973, 2917, 8749, . . .
33. Suppose S is a set with n elements.
(a) How many subsets does S have?
(b) How many ordered pairs of subsets are there?
(c) How many ordered pairs of subsets X, Y satisfy X ∪ Y = S?
(d) How many ordered pairs of subsets X, Y satisfy X ∩ Y = ∅?
(e) How many unordered pairs of subsets X and Y satisfy X ∪Y = S?
(f) How many ordered triples (X, Y, Z) of subsets satisfy X ∪ Y ∪ Z =
S?
(g) How many ordered triples (X, Y, Z) of pairwise disjoint subsets
satisfy X ∪ Y ∪ Z = S?
34. How many five digit numbers abcde satisfy the following two properties:
(a) The digits are all different and all non-zero and
(b) Exactly one of the inequalities a > b, b > c, c > d, and e > d
holds.
35. How many planes in 3-space contain at least three of the vertices of the
unit cube C = [0, 1] × [0, 1] × [0, 1]?
36. How many four-digit numbers have the product of their digits equal to
63?
37. Cam Newton and Steve Smith attend a press conference at the 2012
Pro Bowl with 2 other pairs of NFL teammates. Since the players
spend so much time with their teammates during the season, players
on the same team agree not to sit next to each other. If the six players
sit in a row occupying six consecutive seats, how many ways can the
players sit down for the press conference?
38. In how many rearrangements of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 do the
numbers form a hill ; that is, the numbers form an increasing sequence
at the beginning up to a peak, and then form a decreasing sequence to
the end such as in 129876543 or 258976431, but not 987654321?
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CTI, Nov. 6, 2015
39. Counting Poker Hands
We will be computing the number of five card poker hands that can be
dealt from a standard 52 card deck. In this deck, there are four suits,
(hearts, ♡. diamonds ♢, clubs, ♣, spades, ♠). Each suit contains
a card for each of 13 ranks: 2,3, . . . , 9, 10, Jack, King, Queen, and
Ace. Poker hands are typically characterized by the ranks contained
in the hand. Using combinations and the rule of product, compute the
following:
(a) The total number of 5 card poker hands
(b) The number of 5-card hands which contain at least one card of
each suit
(c) The number of 5-card hands which consist of 4 kings and 1 queen
(d) The number of one pair hands. Of the five cards, that is, 2 are of
the same rank, and the remaining three cards have different ranks
(e.g. 2♠, 2♡, K♣, 8♡, 9♠)
(e) The number of two pairs (e.g. 5♠, 7♠, 3♡, 3♠, 7♣)
(f) The number of three-of-a-kind. Of the 5 cards, 3 have the same
rank and the remaining two cards have different ranks
(e.g. 4♡, 4♠, 4♢, 5♡, 8♣)
(g) The number of straights. The five cards are of sequential ranks in
at least two different suits (4♡, 5♣, 6♡, 7♠, 8♢)
(h) The number of flushes. All five cards hands are of the same suit.
For example, (A♠, K♠, 3♠, 8♠, 9♠). Your answer should include
straight flushes.
(i) The number of full houses. Of the five cards, 3 cards are of
one rank, and the remaining two are of a different rank (e.g.
7♡, 7♠, 7♣, 8♡, 8♢)
(j) The number of four-of-a-kinds. Of the 5 cards, 4 are of one rank,
and the other is of another rank (e.g. A♠, A♢, A♡, A♣, 8♠)
(k) The number of straight flushes. The five cards are all of the same
suit and the ranks are sequential (e.g. 7♢, 8♢, 9♢, 10♢, J♢)
(l) The number of royal flushes, which are straight flushes whose highcard is an ace (e.g. A♠, K♠, Q♠, J♠, 10♠)
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CTI, Nov. 6, 2015
40. How many three-digit numbers are composed of three distinct digits
such that one digit is the average of the other two?
41. How many four digit numbers contain either a 2 or a 3?
42. How many even three-digit integers have the property that their digits,
read left to right, are in strictly increasing order?
43. What is the largest number of regions that a circle can be divided into
by 6 chords of the circle?
44. Distribute n points around the circumference of a circle.
(a) How many chords are determined by the n points?
(b) Assuming no three of the chords in (a) are concentric, how many
points inside the circle belong to two chords.
(c) What is the greatest number of regions the chords in part (a)
could determine.
45. Consider the 3 × 4 grid of lattice points shown below.
•
•
•
•
•
•
•
•
•
•
•
•
(a) How many different slopes can lines connecting two points of the
grid have?
(b) How many triangles have all three vertices in the set?
(c) How many isosceles right triangles have all three vertices in the
set?
(d) How many isosceles triangles have all three vertices in the set?
46. How many ways can 5 A’s, 5 B’s, and 5 C’s be arranged so that there
are no A’s among the first 5, no B’s among the second 5 and no C’s
among the last five?
47. How many subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9} contain at least 1 even
digit?
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Counting Problems
CTI, Nov. 6, 2015
48. Among all the counting numbers we can build using only odd digits,
which is the one thousanth one?
49. Distribute the numbers 1, 2, 3, 6, 7, 8 in the six positions a, b, c, d, e, f
and build the multiplication table.
(a) What is the largest number you can get as the sum of the nine
products ad + ae + af + bd + · · · + cf in the table?
(b) What is the smallest number you can get as the sum of the nine
products ad + ae + af + bd + · · · + cf in the table?
(c) How many different numbers can you get?
(d) Next, answer the same three questions assuming you’re allowed to
build either a 1×5 or a 2×4 table in addition to the 3×3 one above.
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CTI, Nov. 6, 2015
a
b
c
d
ad
bd
cd
e
ae
be
ce
f
af
bf
cf
50. Suppose S = {a1 , a2 , . . . a11 } is a set of eleven points distributed around
a circle. Join each pair of points with a chord. Suppose it turns out
that no point inside the circle belongs to more than 2 such chords.
(a) Find the number of chords.
(b) Find the number of points that belong to two chords.
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Counting Problems
CTI, Nov. 6, 2015
(c) How many triangles have all three of their vertices in the set S?
(d) Find the number of regions inside the circle determined by the
chords.
See the circle below with some, but not all, of the chords drawn.
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a5
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a8•
a•9
51. An increasing arithmetic sequence of length seven is a list a, a + d, a +
2d, . . . , a + 6d where a is a real number and d is a positive real number.
(a) How many increasing arithmetic sequences of length seven contain
both the numbers 90 and 100?
(b) How many increasing arithmetic sequences of length seven contain
the numbers 80, 90 and 100?
(c) How many increasing arithmetic sequences of length seven contain
the numbers 70, 80, 90 and 100?
(d) How many increasing arithmetic sequences of length seven contain
the numbers 60, 70, 80, 90 and 100?
52. Seven lines in the plane, L1 , L2 , . . . , L7 , are given. What is the largest
number k of points that can belong to at least two of the lines? Among
the numbers 0, 1, 2, 3, . . . , k, which numbers can be the number of points
that belong to at least two lines?
16
Counting Problems
CTI, Nov. 6, 2015
53. There are 100 great circles drawn on a sphere so that no three have a
point in common.
(a) How many points belong to two of the circles?
(b) How many regions on the sphere are determined by the great
circles?
54. How many ways can the numbers from 1 to 15 be inserted in the binary
tree of 15 circles so that for every pair of connected circles, the one on
top contains a smaller number.
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17