In mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange.
More formally, a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements, the number of k-combinations is equal to the binomial coefficient
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{\displaystyle {\binom {n}{k}}={\frac {n(n-1)\dotsb (n-k+1)}{k(k-1)\dotsb 1}},}
which can be written using factorials as
{\displaystyle k>n}
. The set of all k-combinations of a set S is often denoted by
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{\displaystyle \textstyle {\binom {S}{k}}}
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Combinations refer to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-selection, k-multiset, or k-combination with repetition are often used. If, in the above example, it were possible to have two of any one kind of fruit there would be 3 more 2-selections: one with two apples, one with two oranges, and one with two pears.
Although the set of three fruits was small enough to write a complete list of combinations, this becomes impractical as the size of the set increases. For example, a poker hand can be described as a 5-combination (k = 5) of cards from a 52 card deck (n = 52). The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter. There are 2,598,960 such combinations, and the chance of drawing any one hand at random is 1 / 2,598,960.
Hello, I've been trying to solve this problem for a while, and I found a technical solution which is too computationally intensive for large numbers, I am trying to solve the problem using Combinatorics instead.
Given a set of integers 1, 2, 3, ..., 50 for example, where R=50 is the maximum...
Summary:: Year 11 Extension 1 Math problem (Australia)
How many combinations can be made from a 4 digit pin code if we can only use two numbers to form our pin code, and we MUST use 2 distinct numbers. E.g. 1112, 4334, 9944, 3232. But NOT 1111, 2113, 0992 etc. We're using the numbers 0-9 and...
Suppose we have two boxes, each containing three types of balls. On each ball there's written a number:
First box: 1, 2, 3
Second box: 4, 5, 6
We don't know how many balls of each type there are, but we know the probability of taking out a specific one, so that we can make a graph showing the...
Homework Statement
There are 15 members of a maths club. There are 4 different medals to be randomly given to the members of the club. What is the probability that no member will receive more than one of the medals.
Homework Equations
Try to find the number of combinations where no member...
Homework Statement
In how many different ways can you seat 11 men and 8 women in a row if no two women are to sit together?
Homework Equations
I have the combination and permutation equations
The Attempt at a Solution
I assume that given the context of this question if I have two, three...
Homework Statement
Question:
A cheese shop carries a large stock of 34 kinds of cheese. By the end of the day 48 cheese sales have been made and the items sold must be restocked. How many different restocking orders are possible?
Homework Equations
Combination and permutation equations
The...
Homework Statement
Say I have four categories which make up a "whole" that I'll call a unique "deal".
Each deal can have "I" properties, "J" investors, "K" mortgages, and "L" credit lines, where "I" and "J" must be integers greater than zero and "K" and "L" are non-negative integers (i.e. 0 or...
Homework Statement
There are 22 students in a class. The professor will divide the class into 4 groups. Group 1 and 2 have 5 members each whilst Group 3 and 4 have 6. Given that the teacher forms the group at random, find the probabilities of :
A = event where Paula, Trina, Gia all belong in...
I have attached an image showing a (what I believe to be) simple problem involving arranging four blocks, each of different dimensions.
Yes, the blocks fit together perfectly in the first arrangement shown in the diagram when there are no gaps.
I'm convinced that the solution is likely easily...
Homework Statement
Homework Equations
The Attempt at a Solution
the answer for no adjoining
_W_W_W_W_W_
for 3 red balls, there are 6 positions
so ## 6C_3 = 20##
i'm curious, on other way to find arrangement?
for adjoining = all arrangement - adjoining
all arrangement = 3 red can get to...
Homework Statement
8 friends are playing a tennis game together. How many different doubles games of tennis can they play?
Homework Equations
Combinations
The Attempt at a Solution
Well, I solved this problem by saying: we choose a group 4 people from 8 to play, so order is not important...
If there was a 1 billion x 1 billion x 1 billion cube made of 3D pixel cubes, and half of them are black and half of them are clear/colorless, then how many combinations of unique pixel arrangements are there?
Would the amount of shapes/objects in this cube be infinite? (Assuming the black...
We have a square grid. In how many ways we get to the point [20,30], if we can not get points [5,5], [15,10] [15,10] [17,23]? The starting point is [0,0], we can only move up and to the right. Thanks
My solutions:
$$ \binom{50}{20} - \binom{10}{5}\binom{40}{15} - \binom{25}{10}\binom{25}{10}...
Can you suggest any book for learning multinomial theorem and its application in permutation and combinations problems? I am also looking for a book for learning Permutations and Combinations. (Right now, I am using a problem oriented book by Marcus. But I want a book for learning the basics as...
Hi!
If I have points A and B in a lattice in the plane, and the closest path between them is n + m steps (for example 4 steps upwards and 5 steps to the right), there are C(9,(5-4)) = 9 combinations of paths between them. I have to choose the 4 ways upwards (or the 5 ways to the right) of the 9...
Hi, I am trying to find the solution (formula) for doing the following:
For any number of players (divisible by three), I need to calculate a number of rounds (in my case 8) for three players to play each other without any repetition.
So the first round of etc. 21 players would be:
1,2,3 4,5,6...
Homework Statement
Consider a 11 digit positive integer formed by the digits 1,0 or both. The probability that no two zeros are adjacent is:
Homework Equations
None
The Attempt at a Solution
First digit has to be 1.
Total number of permutations=210
Now 1XXXXXXXXXX is the format.
Taking 10...
Homework Statement
Is it true that for each ##n\geq 2## there are two primes ##p, q \neq 1## that divide every ##\binom{n}{k}## for ##1\leq k\leq n-1##?
Examples:
For ##n=6: \binom{6}{1}=6; \binom{6}{2}=15; \binom{6}{3}=20; \binom{6}{4}=15; \binom{6}{5}=6.## So we can have ##p=2## and...
I was going through a p and c problem where I had to find the number of non congruent RECTANGLES.
Answer includes number of squares as well.
SHOULD SQUARE BE TAKEN AS A RECTANGLE?