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Combinatorics A: Word Wrod C, O, M, B, o

The document presents a series of combinatorial problems involving distinct words, expected values in random selections, permutations, interesting numbers, cube painting, probabilities in conversations, lattice point colorings, and tournament outcomes. Each problem requires specific calculations or reasoning to derive the desired results. The problems are designed to test knowledge of combinatorics and probability theory.

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allanzhaohm
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18 views1 page

Combinatorics A: Word Wrod C, O, M, B, o

The document presents a series of combinatorial problems involving distinct words, expected values in random selections, permutations, interesting numbers, cube painting, probabilities in conversations, lattice point colorings, and tournament outcomes. Each problem requires specific calculations or reasoning to derive the desired results. The problems are designed to test knowledge of combinatorics and probability theory.

Uploaded by

allanzhaohm
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Combinatorics A

1. [3] A word is an ordered, non-empty sequence of letters, such as word or wrod. How many
distinct words can be made from a subset of the letters c, o, m, b, o, where each letter in the
list is used no more than the number of times it appears?
2. [3] Andrew has 10 balls in a bag, each a different color. He randomly picks a ball from the bag
4 times, with replacement. The expected number of distinct colors among the balls he picks
is pq , where gcd(p, q) = 1 and p, q > 0. What is p + q?

3. [4] Consider a random permutation of the set {1, 2, . . . , 2015}. In other words, for each 1 ≤
i ≤ 2015, i is sent to the element ai where ai ∈ {1, 2, . . . , 2015} and if i 6= j, then ai 6= aj .
What is the expected number of ordered pairs (ai , aj ) with i − j > 155 and ai − aj > 266?
4. [4] A number is interesting if it is a 6-digit integer that contains no zeros, its first 3 digits
are strictly increasing, and its last 3 digits are non-increasing. What is the average of all
interesting numbers?
5. [5] Alice has an orange 3-by-3-by-3 cube, which is comprised of 27 distinguishable, 1-by-1-by-1
cubes. Each small cube was initially orange, but Alice painted 10 of the small cubes completely
black. In how many ways could she have chosen 10 of these smaller cubes to paint black such
that every one of the 27 3-by-1-by-1 sub-blocks of the 3-by-3-by-3 cube contains at least one
small black cube?
6. [6] Every day, Heesu talks to Sally with some probability p. One day, after not talking to Sally
the previous day, Heesu resolves to ask Sally out on a date. From now on, each day, if Heesu
has talked to Sally each of the past four days, then Heesu will ask Sally out on a date. Heesu’s
friend remarked that at this rate, it would take Heesu an expected 2800 days to finally ask
Sally out. Suppose p = m n , where gcd(m, n) = 1 and m, n > 0. What is m + n?

7. [7] The lattice points (i, j) for integers 0 ≤ i, j ≤ 3 are each being painted orange or black.
Suppose a coloring is good if for every set of integers x1 , x2 , y1 , y2 such that 0 ≤ x1 < x2 ≤ 3
and 0 ≤ y1 < y2 ≤ 3, the points (x1 , y1 ), (x1 , y2 ), (x2 , y1 ), (x2 , y2 ) are not all the same color.
How many good colorings are possible?
8. [8] In a tournament with 2015 teams, each team plays every other team exactly once and no
ties occur. Such a tournament is imbalanced if for every group of 6 teams, there exists either
a team that wins against the other 5 or a team that loses to the other 5. If the teams are
indistinguishable, what is the number of distinct imbalanced tournaments that can occur?

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