Articles of permutations

A team of 11 players with at least 3 bowlers and 1 wicket keeper is to be formed from 16 players; 4 are bowlers; 2 are wicket keepers.

Out of 16 players in a cricket team, 4 are bowlers and 2 are wicket keepers. A team of 11 players with at least 3 bowlers and 1 wicket keeper is to be formed. Find the number of ways the team can be selected. My solution: Choosing 3 bowlers out of 4: $\binom43$ Choosing 1 […]

Balls in bins, probability that exactly two bins are empty

If we throw randomly $n+1$ balls into $n$ bins, what is the probability that exactly two bins are empty? I tried to do this problem but I didn’t get right solution, I would be very grateful if someone would point where am I making mistake, and how should I think about that. Also, if there […]

How many ways to put 20 things to different 4 boxes?

I have 20 identical balls. I want to put these 20 balls to 4 different boxes. In how many ways I can do it? (If necessary we can keep one or more boxes empty)

Combinatorics: How to find the number of sets of numbers in increasing order?

The problem is the following one: Let $n$ and $m$ be natural numbers and $m < n$. Find $m$-permutations of the set $\{1, 2,\dots, n\}$ such that permutations are in non-decreasing order (for both cases where repetition is allowed and where it is not allowed). Tried to solve, still have no ideas. Thanks in advance.

2 regular graphs and permutations

I have found this question on MSE before but I didn’t find the answer satisfactory and it is so old I doubt anyone is still following it. Let $f_{n}$ be the number of permutations on $[n]$ with no fixed points or two cycles. Let $g_{n}$ be the number of simple, labeled two regular graphs on […]

How many seven-digit numbers satisfy the following conditions?

How many seven-digit numbers divisible by 11 have the sum of their digits equal to 59? I am able to get the seven-digit numbers divisible by 11 and I am also able to get the seven-digit numbers whose sum of their digits equal to 59. But i am not able to get how i can […]

How to geometrically show that there are $3$ $D_4$ subgroups in $S_4$?

As shown in this note, the symmetry group $S_4$ for a cube has $3$ subgroups that are isomorphic to $D_4$, the dihedral group of order $2 \times 4 = 8$. How to geometrically illustrate this fact? Specifically, where are the squares embedded in the cube? Related post: How to geometrically show that there are $4$ […]

Writing the identity permutation as a product of transpositions

I am reading Introduction to Abstract Algebra by Keith Nicholson and ran into a lemma that states: If the identity permutation $\varepsilon$ can be written as a product of $n \geq 3$ transpositions, then it can be written as a product of $n – 2$ transpositions. So let us look at $S_4$. So $\varepsilon = […]

What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters?

What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters? I believe the answer for this is 6. As we can write the group elements as below (a)(b)(C) (ab)(c) (ac)(b) (bc)(a) (abc) (bac) Can we generalize that for any $S_n$ onto $n$ the number of automorphisms will be $n!$ Also […]

Unique intermediate subgroup and double coset relation I

Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$). Question: Is it true that $HgK=KgH$, $\forall g \in G$ ? Experiment (GAP) : It’s true if $[G:H] \le 30$ and $\vert G/H_G \vert \le 10000$. ($H_G$ is […]