## Probability combinatorics deck of cards

When choosing 13 cards from a deck of 52 cards, what is the probability of choosing precisely 6,7,8 of the same suit?

## combinatorics – Is there a way to list all partitions of a set by moving one element at a time?

Given a set $$S = {1, 2, ldots, n}$$, imagine that you start from an initial partition $$P^{(0)} = {P^{(0)}_1, P^{(0)}_2, ldots}$$ of $$S$$, then you a create new one $$P^{(1)}$$ by moving only one element from some subset $$P^{(0)}_i$$ to some subset $$P^{(0)}_j$$, and so on. Is it possible to list all partitions of $$S$$ in this way, such that no partition repeats in the list?

It’s possible when $$n=1,2,3$$. I don’t know for $$n=4$$. Any help is appreciated!

## combinatorics – What is the probability of observing unique colors in a combinatorial problem?

Suppose I have $$K$$ colors and for each color there are $$N$$ balls, so there are $$K*N$$ balls in total. Now I draw $$M$$ balls from them. For these $$M$$ balls, some balls have the same color and some don’t. Let $$K_u$$ be the number of colors that are unique in the $$M$$ balls. My question is: what is the expectation of $$K_u/K$$?

## combinatorics – What counting problem would have this solution?

I need to create a counting problem that has the following formula as a solution, where $$n$$ and $$k$$ are positive integers:

$${n choose k } 2^k(n-k)_k$$ where $$(n)_i = n(n-1)(n-2)…(n-i+1)$$ is the falling factorial.

My theoretical problem that would have this solution is one of ordering beads. Suppose we have an inexhaustible amount of black beads, white beads, and $$n−k$$ other colors of beads (red, blue, green, etc., etc.) (By “inexhaustible” it is meant that “at least $$k$$ black beads”, “$$k$$ white beads”, and “at least one bead of each of the $$n−k$$ other colors”.)

How many ways/orders can we pick n beads such that exactly $$k$$ beads are black or white and the remaining $$n−k$$ beads have no repeated colors?

My understanding is we are taking beads and lining them up in a certain order and finding the total different ways. The solution would then be $${n choose k}2^k(n-k)_k$$ is this true?

## combinatorics – Alternative way to calculate number of edges in Turán-graphs?

We define a Turán-graph
$$T_n(r) = K_{lceilfrac{n}{r}rceil, ldots,lceilfrac{n}{r}rceil, lfloorfrac{n}{r}rfloor, ldots, lfloorfrac{n}{r}rfloor}$$

with $$n bmod r$$ many subsets $$V_i$$ that contain $$lceilfrac{n}{r}rceil$$ many vertices and $$r- (n bmod r)$$ many $$lfloorfrac{n}{r}rfloor$$ vertices. We calculate the number of edges in such a graph as follows $$t_n(r) = left( 1 – frac{1}{r}right)frac{n^2}{2} – frac{q(r-q)}{2r}$$

## My thoughts:

We can think of Turán-graphs as $$r$$-partit graphs. So for some $$T_n(r) = K_{V_1, V_2, ldots, V_r}$$ with $$e_{i} = binom{i}{2}$$ representing the number of edges in some complete graph $$K_i$$.

Hence, we can calculate the number of edges as follows:
begin{align*} t_n'(r) &= e_{n} – e_{V_1} – e_{V_2} – ldots – e_{V_r} \ &= binom{n}{2} – binom{V_1}{2} – binom{V_2}{2} – ldots -binom{V_r}{2}end{align*}

## Question:

This seems way more simple to me than the formula given above. If $$t_n'(r)$$ works correct (couldn’t prove it wrong nor right), why bother using $$t_n(r)$$?

## Combinatorics optimization problem – Mathematics Stack Exchange

Intro: As a biotechnologist I’m not very proficient with math and would need some help. Please forgive my lack of mathematical annotations.

Problem: I have a large set of 1D-arrays of size 4 and value range [0, 200]. E.g.: [0, 10, 83, 92], [20, 180, 10, 100], [1, 0, 9, 2], [150, 20, 30, 20], etc.
I would like to combine/group these arrays such that the addition of the arrays in one group does not exceed [200, 200, 200, 200]. Furthermore, the number of groups should be minimized, making it a combinatorics and optimization problem.

Example: E.g. one group can contain [50, 50, 50, 50] + [50, 50, 50, 50] + [100, 100, 100, 100] = [200, 200, 200, 200], but not [180, 180, 180, 180] + [100, 100, 100, 100] > 200.

Please ask questions if anything remains unclear. Thank you in advance and kind regards.

## combinatorics – Let \$Bsubset A = {1,2,3,…,99,100}\$ and \$|B|= 48\$. Prove that exist \$x,yin B\$, \$xne y\$ such that \$11mid x+y\$.

Let $$Bsubset A = {1,2,3,…,99,100}$$ and $$|B|= 48$$. Prove that
exist $$x,yin B$$, $$xne y$$ such that $$11mid x+y$$.

Proof: Let $$P_0= {11,22,…,99}$$ and for $$i= 1,2,…49$$ and $$11nmid i$$ make pairs $$P_i:= {i,99-i}$$. Now we
have $$46$$ subsets with sum of each pair in each subset divisible
by 11. So if we took at most 1 element from each set in to $$B$$
then $$B$$ would have at most 47 elements (if $$100$$ is in $$B$$ also).
A contradiction.

• Now can this bound be sharpened down. I can not find
counterexample with $$|B|leq 47$$.
• Also, is this problem doable by polynomial method? As to watch in $$mathbb{Z}_{11}(x,y)$$ a polynomial $$p(x,y) = prod_{i=1}^{10}(x+y-i)$$

## combinatorics – How many ways to put balls in boxes with restrictions?

Suppose we had eight unique and labeled boxes (1-8) and 16 indistinguishable orange balls. How many ways to arrange:

• 1) If odd boxes must have an odd number of balls and even boxes have an even number of balls?
• 2) If we also have 16 indistinguishable yellow balls and want to distribute both orange and yellow?

I am kind of clueless on where to go from the first part. For the second part, would we just add the balls together such that we have $$32$$ balls? Then $$dfrac{39!}{7!times32!}$$. Thanks!

## combinatorics – Chromatic polynomial of the cross-polytope and denominators of convergents to e.

Let $$C_n$$ denote the $$1$$-skeleton of the $$n$$-dimensional cross-polytope, and $$chi_{C_n}(x)$$ be the chromatic polynomial of $$C_n$$. This is equivalent to the way of coloring the $$(n-1)$$-dimensional faces of the $$n$$-dimensional hypercube with $$x$$ colors so that if two of them share a $$(n-2)$$-dimensional face, they have different colors.

Then

• $$chi_{C_1}(x) = x^2,$$
• $$chi_{C_2}(x) = x(x-1)(x^2-3x+3),$$
• $$chi_{C_3}(x) = x(x-1)(x-2)(x^3 – 9x^2 + 29x – 32),$$
• $$chi_{C_4}(x) = x(x-1)(x-2)(x-3)(x^4 – 18 x^3 + 125 x^2 – 392 x + 465),$$
• $$chi_{C_5}(x) = x(x – 1) (x – 2)(x – 3) (x – 4)(x^5 – 30 x^4 + 365 x^3 – 2240 x^2 + 6909 x – 8544),$$ and more generally,
• $$displaystyle chi_{C_n}(x) = left(prod_{i=0}^{n-1}(x-i)right)f_n(x)$$.

It appears that the absolute value of the constant term of $$f_n(x)$$ is equal to OEIS sequence $$A007677(3n – 4)$$ for $$n geq 2$$, where A007677 is a list of “denominators of convergents to $$e$$“.

Is this a coincidence, or is there a connection between hypercubes and convergents of $$e$$?

I’m surprised by this. Should I be?

## combinatorics – Combinations (repetition not allowed & order not important)

How to compute a table of numbers (all possibilities), where repetition is not allowed and order is not important.
Example:

I have a set of prime numbers. In this example I have four: {3,5,7,11}, but it can be anything, and I want to choose one pair(two elements) out of that set.
To make things easier, I want to compute the indices to get those pairs of prime numbers. The set of indices is then {0,1,2,3}. We pick 2 out of 4 elements. So how do we compute the permutations or combinations:

``````0,1   (3,5)
0,2   (3,7)
0,3   (3,11)
1,2   (5,7)
1,3   (5,11)
2,3   (7,11)
``````

?

It was difficult to find examples on the web, because they either allowed repetitions or were order was important. Pls answer with pseudocode or c/c++ if you can.