Questions tagged [combinations]

Combinations are subsets of a given size of a given finite set. All questions for this tag have to directly involve combinations; if instead the question is about binomial coefficients, use that tag.

A combination is a way of choosing elements from a set in which order does not matter.

A wide variety of counting problems can be cast in terms of the simple concept of combinations, therefore, this topic serves as a building block in solving a wide range of problems.

The number of combinations is the number of ways in which we can select a group of objects from a set.

The difference between combinations and permutations is ordering. With permutations we care about the order of the elements, whereas with combinations we don’t.

Notation: Suppose we want to choose $~r~$ objects from $~n~$ objects, then the number of combinations of $~k~$ objects chosen from $~n~$ objects is denoted by $~n \choose r~$ or, $~_nC_r~$ or, $~^nC_r~$ or, $~C(n,~r)~$.

$~n \choose r~$$=\frac{1}{r!}~^nP_r=\frac{n!}{r!~(n-r)!}$

Example: Picking a team of $~3~$ people from a group of $$~10\cdot C(10,3) = \frac{10!}{7! \cdot 3!} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120.~$$

7125 questions
69
votes
18 answers

Proof of the hockey stick/Zhu Shijie identity $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$

After reading this question, the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$ What's the name of this identity? Is it the identity of the Pascal's triangle modified. How can we prove it? I tried by…
hlapointe
  • 1,530
  • 1
  • 14
  • 26
59
votes
9 answers

How many 7-note musical scales are possible within the 12-note system?

This combinatorial question has a musical motivation, which I provide below using as little musical jargon as I can. But first, I'll present a purely mathematical formulation for those not interested in the motivation: Define a signature as a…
MGA
  • 9,204
  • 3
  • 39
  • 55
44
votes
0 answers

What Rubik's Twist configuration has the lowest visible surface area?

The Rubik's Twist has been a fun time sink. From the wiki page, [It] is a toy with twenty-four wedges that are right isosceles triangular prisms. The wedges are connected by spring bolts, so that they can be twisted, but not separated. By being…
Elle
  • 491
  • 3
  • 8
41
votes
6 answers

The pigeonhole principle and a professor who knows $9$ jokes and tells $3$ jokes per lecture

A professor knows $9$ jokes and tells $3$ jokes per lecture. Prove that in a course of $13$ lectures there is going to be a pair of jokes that will be told together in at least $2$ lectures. I've started with counting how many possibilities…
user565804
34
votes
7 answers

How many lists of 100 numbers (1 to 10 only) add to 700?

Each number is from one to ten inclusive only. There are $100$ numbers in the ordered list. The total must be $700$. How many such lists? Note: if, as it happens, this is one of those math problems where only an approximation is known, that would…
Fattie
  • 1,360
  • 11
  • 19
28
votes
13 answers

Calculating the number of possible paths through some squares

I'm prepping for the GRE. Would appreciate if someone could explain the right way to solve this problem. It seems simple to me but the site where I found this problem says I'm wrong but doesn't explain their answer. So here is the problem…
user120865
  • 289
  • 1
  • 3
  • 4
25
votes
5 answers

Passwords: Two 50-characters vs one 100-characters

In this Information Security question, we discuss whether or not a $100$ character secret randomly-generated username is equivalent to a $50$ character secret randomly-generated username plus a $50$ character secret randomly-generated password. This…
24
votes
4 answers

Formula for Combinations With Replacement

I understand how combinations and permutations work (without replacement). I also see why a permutation of $n$ elements ordered $k$ at a time (with replacement) is equal to $n^{k}$. Through some browsing I've found that the number of combinations…
Xoque55
  • 3,965
  • 3
  • 21
  • 47
24
votes
1 answer

Lottery Math (different combinations)

In my country, Brazil, we have a lottery game called "Mega-Sena". You can choose from 6 (cheapest set) to 15 (most expensive set) numbers from a total of 60. *Blue: Chosen numbers; *Green: Amount of chosen numbers. Every week they have a new…
Lucas NN
  • 352
  • 2
  • 8
22
votes
8 answers

Number of ways to connect sets of $k$ dots in a perfect $n$-gon

Let $Q(n,k)$ be the number of ways in which we can connect sets of $k$ vertices in a given perfect $n$-gon such that no two lines intersect at the interior of the $n$-gon and no vertex remains isolated. Intersection of the lines outside the $n$-gon…
Matan
  • 791
  • 4
  • 12
22
votes
5 answers

In how many ways can $1000000$ be expressed as a product of five distinct positive integers?

I'm trying to solve the following problem: "In how many ways can the number $1000000$ be expressed as a product of five distinct positive integers?" Here is my attempt: Since $1000000 = 2^6 \cdot 5^6$, each of its divisors has the form $2^a \cdot…
user144765
  • 691
  • 1
  • 5
  • 17
21
votes
4 answers

How many triangles can be formed by the vertices of a regular polygon of $n$ sides?

How many triangles can be formed by the vertices of a regular polygon of $n$ sides? And how many if no side of the polygon is to be a side of any triangle ? I have no idea where I should start to think. Can anyone give me some insight ? Use…
Bsonjin
  • 315
  • 1
  • 3
  • 8
19
votes
5 answers

Probability of two people meeting in a given square grid.

Amy will walk south and east along the grid of streets shown. At the same time and at the same pace, Binh will walk north and west. The two people are walking in the same speed. What is the probability that they will meet? I tried using Pascal's…
Sarhad Salam
  • 325
  • 2
  • 8
19
votes
1 answer

Hard combinatorics and probability question.

A large white cube is painted red, and then cut into $27$ identical smaller cubes. These smaller cubes are shuffled randomly. A blind man (who also cannot feel the paint) reassembles the small cubes into a large one. What is the probability that the…
Aditya Kumar
  • 1,459
  • 11
  • 23
16
votes
2 answers

Counting the number of triangles inside $3-4-5-$triangle [Found in Arabic Math book: الرياضيات | هندسة الإحداثيات | الإحصاء]

While reading a pdf Arabic math book, counting chapter, I found this question: It says: The points $(0,0),(0,3),(4,0)$ are jointed to each other. Also, the points: $(0,1),(0,2),(0.8,2.4),(1,0),(1.6,1.8),(2,0),(2.4,1.2),(3,0),(3.2,0.6)$ are…
1
2 3
99 100