| 01 |
Arranging MISSISSIPPI
How many distinct arrangements are there of the letters in MISSISSIPPI?
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Combinatorics |
3.0/10
|
Unsolved
|
| 02 |
Birthday Paradox
How many people need to be in a room before there is at least a 50% chance that two of them share a birthday?
|
Combinatorics |
4.0/10
|
Unsolved
|
| 03 |
Counting Surjections
How many functions from a 5-element set onto a 3-element set hit every element of the codomain?
|
Combinatorics |
2.6/10
|
Unsolved
|
| 04 |
Derangements
n letters are placed randomly into n addressed envelopes. As n → ∞, what does the probability that no letter ends up in its correct envelope approach?
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Combinatorics |
5.0/10
|
Unsolved
|
| 05 |
Eight Rooks
In how many ways can 8 rooks be placed on a chessboard so that no two attack each other?
|
Combinatorics |
3.0/10
|
Unsolved
|
| 06 |
Grid Paths
How many right/up paths cross an 8 × 8 grid from corner to corner?
|
Combinatorics |
3.0/10
|
Unsolved
|
| 07 |
Guaranteed Pair Sum
How many numbers must you pick from {1, ..., 100} to guarantee two of them sum to 101?
|
Combinatorics |
4.0/10
|
Unsolved
|
| 08 |
Making Change
In how many ways can you make 25 cents from pennies, nickels, and dimes?
|
Combinatorics |
4.0/10
|
Unsolved
|
| 09 |
Moser's Circle
Join six points on a circle with all chords. The region counts 1, 2, 4, 8, 16 suggest 32 next; what is it really?
|
Combinatorics |
5.0/10
|
Unsolved
|
| 10 |
No Consecutive Elements
How many subsets of {1, ..., 10} contain no two consecutive integers?
|
Combinatorics |
4.0/10
|
Unsolved
|
| 11 |
Partitioning a Set
In how many ways can a 5-element set be split into any number of nonempty unlabeled blocks?
|
Combinatorics |
7.0/10
|
Unsolved
|
| 12 |
Paths Below the Diagonal
How many lattice paths from (0,0) to (10,10) never rise above the diagonal? The answer is a famous sequence in disguise.
|
Combinatorics |
3.0/10
|
Unsolved
|
| 13 |
Polygon Diagonals
How many diagonals does a convex 12-sided polygon have?
|
Combinatorics |
3.0/10
|
Unsolved
|
| 14 |
Stars and Bars
How many nonnegative integer solutions does x₁ + x₂ + x₃ + x₄ = 10 have?
|
Combinatorics |
3.6/10
|
Unsolved
|
| 15 |
The Ballot Problem
Candidate A gets 60 votes, Candidate B gets 40. What is the probability A was strictly ahead throughout the entire count?
|
Combinatorics |
4.5/10
|
Unsolved
|
| 16 |
The Handshake Count
At a party of 20 people, everyone shakes hands with everyone else exactly once. How many handshakes?
|
Combinatorics |
3.0/10
|
Unsolved
|
| 17 |
The Josephus Survivor
100 people stand in a circle and every second person is eliminated until one remains. Which position survives?
|
Combinatorics |
5.0/10
|
Unsolved
|
| 18 |
The Mutilated Chessboard
Remove two opposite corners from a chessboard. Can 31 dominoes tile the remaining 62 squares?
|
Combinatorics |
5.0/10
|
Unsolved
|
| 19 |
The Round Table
In how many distinct ways can 8 people sit around a circular table, counting rotations as the same?
|
Combinatorics |
4.0/10
|
Unsolved
|
| 20 |
Threes and Fives
How many integers from 1 to 1000 are divisible by 3 or 5?
|
Combinatorics |
2.2/10
|
Unsolved
|
| 21 |
Tiling with Dominoes
How many ways can a 2 × 10 board be tiled with 1 × 2 dominoes?
|
Combinatorics |
3.0/10
|
Unsolved
|
| 22 |
Trailing Zeros
How many zeros does 100! end in?
|
Combinatorics |
3.0/10
|
Unsolved
|
| 23 |
Two-Color Necklaces
How many distinct necklaces of 6 black-or-white beads exist, counting rotations as the same?
|
Combinatorics |
3.3/10
|
Unsolved
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