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Combinations of Poker Hands [05/15/1998]

Counting three of a kind, two pair, and one pair poker hands.

Combinations of Prisoners [09/02/1997]

Nine prisoners are taken for their daily exercise handcuffed together in threes. How would the warden arrange the men each day so that no two men are handcuffed together more than once over a six day period?

Combinations of Three Words [8/28/1996]

I have three columns of 20 words each...

Combinations of Toppings when Ordering a Pizza [05/19/2005]

I was working on calculating how many combinations of pizza can be made using 6 different toppings with no double toppings allowed, and I found that both c(6,0) + c(6,1) + c(6,2) + ... + c(6,6) and 2^6 gave me the same correct answer of 64. Why do both methods work?

Combinations of X's and Y's [10/27/1999]

X's and Y's are written in a row (e.g. XX...XXYY...Y). How many different arrangements of the letters can there be?

Combinations Totaling 100 [09/27/1999]

In how many ways can I achieve a sum of 100 adding together only 6 integers taken from the set of integers from 1 to 44?

Combinations with Duplicate Objects [06/22/1999]

How many different combinations are there when choosing 3 letters from the group {ABBCCC}?

Combinatorial Proof [6/13/1996]

Please prove this combinatorial proof.

Combinatorial Proof [7/16/1996]

How do I prove that C(n,r)C(r,k) = C(n,k)C(n-k,r- k) where k <= r <= n?

Combinatorial Proof: Identity [02/10/2003]

Prove: C(2n,4) = 2C(n,4) + 2C(n,3)C(n,1) + (C(n,2))^2 for all n in N; n greater than or equal to 4.

Combinatorics [08/19/1997]

sum((-1)^k*((4n choose 2k)/(2n choose k)) for k = 1 to 2n

Combinatorics in a 4x4 Board with White and Black Tiles [07/27/2006]

8 white and 8 black tiles are arranged in a 4x4 square such that in each row and each column there are 2 white and 2 black tiles. In how many different patterns can the tiles be arranged?

Combinatorics: Ramsey Theory [01/12/1998]

Could you help me with a detailed explanation of the theory and a concrete example?

Combinatorics: Unique Groupings [5/30/1996]

Twenty-four friends want to play as many rounds of golf as they can... how many unique rounds of golf can they play?

Comparing Corresponding Factors [03/04/2003]

Show that C(p-1,k) = ((p-1)*...*(p-k))/(1*...*k) is congruent to (-1)^ k mod p.

Connect 4 - Number of Winning Arrangements [6/20/1996]

Can you help us find the formula for the number of winning lines on a 7 x 6 Connect-4 board?

Connecting the Dots [03/14/1999]

If you have a few dots on a page, how many lines does it take to connect them all to each other?

Counterfeit Coin Challenge [05/12/2007]

In a set of 13 coins, either zero or two of them are counterfeit and are lighter. You must identify the counterfeit coins, if any, after four or fewer weighings on a balance scale.

Counting Alternating Subsets [03/29/2003]

Find the number of alternating subsets of A_n.

Counting Arrangements of Objects in a Set [01/17/2006]

How many different ways can you arrange the elements in a set? Two elements (a and b) can be ab or ba, but it's harder with bigger sets. An explanation of the math underlying the Multiplication Principle.

Counting Digits [10/23/1998]

Using the Fundamental Principle of Counting, how many six-digit numbers can you make with two zeros, two twos, and two fours?

Counting Handshakes [9/7/1995]

There are 20 people in a room, and each person shakes hands just once with everyone else. Wouldn't this be 380 total handshakes? 19 * 20 = 380.

Counting Intersections of Diagonals in Polygons [03/08/2000]

Can you help me find an equation for the maximum number of intersections of the diagonals in a polygon?

Counting Paths on a Grid [08/21/2006]

On a 4 x 5 grid, how many possible routes are there to go from one corner to the diagonally opposite corner?

Counting Paths With Factorials [07/24/2002]

In the diagram, how can I find the number of paths from point A to point B, using factorials?

Counting Patterns in Stacking Bricks [01/04/2009]

I have a number of bricks which are each 3 units long, 1 unit deep and 1 unit wide. I want to stack them in a tower 3 units wide, 1 unit deep and 10 units high. How many ways can that be done?

Counting Possible Combinations of Weights [10/07/2004]

If you have a set of weights, in sizes 50g, 25g, 15g, and 5g, how would you go about determining how many possible combinations of weights you could make to equal 85g?

Counting Possible Letter Arrangements [10/25/2004]

In rearranging the letters of the word GUMTREE, how many ways can the letter M be to the left of the two E's?

Counting Possible Paths [05/30/2002]

How do I find the number of pathways from A to B?

Counting Rectangles [05/23/2001]

How can I find the number of different rectangles in a square grid containing "c" columns and "r" rows?

Counting Squares in Bigger Squares [02/29/2000]

How many edge 2 squares (2x2 squares) can be found in an edge 4 square (a 4x4 square)?

Counting Triangles [05/27/1999]

In a large triangle with 36 small ones inside, how many triangles are there in all?

Creating a List of Permutations with a Computer Program [07/31/2007]

I'm writing a computer program to list all the permutations of a given set of numbers, but am not sure of the best way to do it. Any ideas?

Creating an Organized List of Combinations [11/25/2008]

I know how to use the formula to determine the number of combinations of 6 numbers drawn from 12 numbers (1 to 12 inclusive). But without the formula, how can I make a list of all the possible combinations and be confident that I've found them all?

Cuisenaire Rod Combinations [11/13/2001]

We are to record 512 different combinations of each level (or color) of rod, with the orange rod (the largest) being the last. Is there a formula exist to find these combinations without literally manipulating the rods?

Darts Tournament with Eight Players [10/05/2002]

There are eight players in a darts tournament. Each player plays one game against each of the other players. How many dart games will be played in the tournament?

Data Compression [08/10/2003]

Is there an algorithm that can reduce any binary number to a much smaller binary number, then later be reversed to regain the original number? It has to work for any binary number.

Dealing with Duplicate Elements [05/17/2002]

I need to develop a general formula to calculate the number of different 'words' that can be formed from an N letter word where some of the letters may be duplicated.

Digits that Average [7/19/1996]

How many integers from 100 to 999 inclusive have one digit that is the average of the other two?

Dinner Triplets [10/23/2000]

A woman has 15 friends. For 35 days she wants to have dinner with 3 friends a day, arranging it so that each pair of friends will come only once. Is this possible?

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