Casework I
Daniel Kim
(1)
Tina randomly selects two distinct numbers from the set
,
and Sergio randomly selects a number from the set
. What is
the probability that Sergio's number is larger than the sum of the two
numbers chosen by Tina?
(2)
(3)
Coin
is flipped three times and coin
is flipped four times. What is the
probability that the number of heads obtained from flipping the two fair
coins is the same?
How many three-digit numbers satisfy the property that the middle digit is
the average of the first and the last digits?
(Hint: There is a smarter constructive counting way with some casework)
(4)
(5)
Three red beads, two white beads, and one blue bead are placed in line in
random order. What is the probability that no two neighboring beads are the
same color?
Bernardo randomly picks 3 distinct numbers from the set
and arranges them in descending order to form a
3-digit number. Silvia randomly picks 3 distinct numbers from the set
and also arranges them in descending order to form
a 3-digit number. What is the probability that Bernardo's number is larger
than Silvia's number?
(6)
(7)
(8)
(9)
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts.
Some, but not all, of them are internet friends with each other, and none of
them has an internet friend outside this group. Each of them has the same
number of internet friends. In how many different ways can this happen?
Amy, Beth, and Jo listen to four different songs and discuss which ones they
like. No song is liked by all three. Furthermore, for each of the three pairs of
the girls, there is at least one song liked by those two girls but disliked by
the third. In how many different ways is this possible?
How many three-digit numbers are not divisible by , have digits that sum to
less than
, and have the first digit equal to the third digit?
A positive integer is nice if there is a positive integer
with exactly four
positive divisors (including and ) such that the sum of the four divisors
is equal to . How many numbers in the set
are nice?
(10)
(11)
(12)
Four fair six-sided dice are rolled. What is the probability that at least three
of the four dice show the same value?
Eight people are sitting around a circular table, each holding a fair coin. All
eight people flip their coins and those who flip heads stand while those who
flip tails remain seated. What is the probability that no two adjacent people
will stand?
Al, Bill, and Cal will each randomly be assigned a whole number from to
, inclusive, with no two of them getting the same number. What is the
probability that Al's number will be a whole number multiple of Bill's and
Bill's number will be a whole number multiple of Cal's?
(13)
A rectangular box measures
, where , , and are integers and
. The volume and the surface area of the box are
numerically equal. How many ordered triples
(14)
are possible?
There are
people standing equally spaced around a circle. Each person
knows exactly of the other people: the people standing next to her or
him, as well as the person directly across the circle. How many ways are
there for the
people to split up into pairs so that the members of each
pair know each other?
SOURCE:
(1) https://artofproblemsolving.com/wiki/index.php/2002_AMC_12A_Problems/Problem_16
(2) https://artofproblemsolving.com/wiki/index.php/2004_AMC_10A_Problems/Problem_10
(3) https://artofproblemsolving.com/wiki/index.php/2005_AMC_10A_Problems/Problem_14
(4) https://artofproblemsolving.com/wiki/index.php/2008_AMC_10B_Problems/Problem_22
(5) https://artofproblemsolving.com/wiki/index.php/2010_AMC_12A_Problems/Problem_16
(6) https://artofproblemsolving.com/wiki/index.php/2012_AMC_10A_Problems/Problem_23
(7) https://artofproblemsolving.com/wiki/index.php/2012_AMC_12B_Problems/Problem_16
(8) https://artofproblemsolving.com/wiki/index.php/2013_AMC_10A_Problems/Problem_13
(9) https://artofproblemsolving.com/wiki/index.php/2013_AMC_10B_Problems/Problem_24
(10) https://artofproblemsolving.com/wiki/index.php/2014_AMC_10B_Problems/Problem_
16
(11)
https://artofproblemsolving.com/wiki/index.php/2015_AMC_10A_Problems/Problem_
22
(12) https://artofproblemsolving.com/wiki/index.php/2015_AMC_10B_Problems/Problem_
16
(13) https://artofproblemsolving.com/wiki/index.php/2015_AMC_10B_Problems/Problem_
25
(14) https://artofproblemsolving.com/wiki/index.php/2020_AMC_10B_Problems/Problem_
17