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.