Combination Formula

Source: Khan Academy

To know the combination formula, it is important to know that what is a combination first. A combination is a selection of some or all of several different objects from a collection where the order of selection does not matter or is simply immaterial. The number of selections of r objects from the given n objects is denoted by nCr and is given by

nCr = n! / r!(n−r)! 

Combinations make reference to the mix of n things taken k at a time without repetition and where the order of selection does not matter and only the selection of certain element matters.

The combination is simply the way the number of ways we can select a group of objects from a set of objects.

If the problem calls for the number of ways of selecting objects and the order of selection is not to be counted, then the combination can be used. There are two types of combination. One where repetition is allowed and one where repetition is not allowed.

  • Combination-selection of objects
  • Order does not matter
  • nCdenotes combination
  • Can be with repetition
  • Can be without repetition

Check Out: Permutation and Combination- Prepare for the Best Result


How To Solve Combination Formula?

The formula for solving combination problems is nCr = n! / r!(n−r)! . This formula is applicable to combinations where repetition is not allowed.

Here in the above formula where n is the number of things to choose from, and r is the things we choose from n of them, without repetition and order does not matter.

Simply insert the number the values in the formula and reduce it by factorials. The factorial can be solved by multiplying all whole numbers from our chosen number down to 1. Solve the factorial till the answer is reduced 

Combination Formula With Example

The combination formula is nCr = n! / r!(n−r)! , this formula applies to situations where there is no repetition of objects selected.

Where there is repetition allowed formula is changed to (r+n-1)!/r! (n-r)! where n is the number of things to choose from, and we choose r of them and repetition allowed and order does not matter.

Q. Find the number of subsets of the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} having 4 elements.
Answer: The set has 11 elements. Any subset that we form has to have 4 elements from the set. Here the order of choosing the elements doesn’t matter. The set { 1, 2, 3, 4} is the same as {4, 3, 2, 1}. Therefore, this is a problem in combinations. Thus we have to find the number of ways of choosing 4 numbers of this set which has 11 elements.

We can do this by using the formula for combinations as:
11 C 4 = 11!/4!(11-4)! = 11!/7! = (11.10.9.8)/4.3.2.1 = 330 ways. Hence the correct option is C) 330.

Q. In how many ways can 21 books on English and 19 books on Hindi be placed in a row on a shelf so that no two books on Hindi may not be together?
Answer. No two Hindi books are together.
Number of English books = 21
Number of Hindi books = 19
Hindi books can be placed in the gaps between English books.
Since the total number of English books = 21, It can be placed in 22 gaps including the ends.
Hence the number of combinations is given by =
22C19
We know that nCr = n! / [(n – r)! * r!]
22C19= (22*21*20*19! )/(19! * 3! )
= (22*21*20)/6
= 1540 ways.

Q. At an election there are 5 candidates and 3 members are to be elected. A voter is entitled to vote for any number of candidates not greater than the number to be elected. In how many ways a voter can vote?
Answer. A voter can give either 1 vote, 2 votes or 3 votes.
Number of ways to give only 1 vote = 5C1 = 5 
Number of ways to give only 2 vote = 5C2 = 10 
Number of ways to give all 3 vote = 5C3 = 10 
so, a voter can cast his vote by total : 5+10+10 = 25 ways

Q. A box contains three white balls, four black balls, and three red balls. The number of ways in which three balls can be drawn from the box so that at least one of the balls is black is?
Answer. The required number of ways
(a) 1 black and 2 others = 4C1.6C2 = 4 × 15 = 60
(b) 2 black and 1 other = 4C2.6C1 = 6 × 6 = 36
(c) All the three black = 4C3 = 4
Total =60 + 36 + 4 = 100


Check Out: Mensuration


Combination Formula Calculator

Over the internet, there are many websites that help you to find the number of combinations from a set of objects. The Online Combination formula calculator is a simple combination calculator. Users just need to enter the n and r in the fields and it will calculate the combination. This saves time and no need for manual calculation is required.

  • The combination can be calculated online also
  • Combination calculator websites help to find a combination
  • These websites are easy to use 
  • Saves time and reduces calculation 

Combination Formula In Excel

The combination can also be calculated in MS Excel. Microsoft Excel has the COMBIN function that returns the number of combinations for a specified number of items. The COMBIN formula is as follows – COMBIN( number, chosen )

Where number- The number of total items and chosen is the number of items in a combination that are chosen. The COMBIN function will always return a numeric value.

  • MS Excel also provides a function to solve combination
  • COMBIN function is used
  • The answer is a numeric value
  • It is simple to use

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Combination Formula Probability

Permutations and combinations are used to solve advanced and complex probability problems.  Thus to calculate the number of total outcomes, one might have to calculate combinations for the same. 

Example- A 4 digit PIN is selected. What is the probability that there are no repeated digits?
There are 10 possible values for each digit of the PIN (namely: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9), so there are 10 × 10 × 10 × 10 = 10
4 = 10000 total possible PINs.

To have no repeated digits, all four digits would have to be different, which is selecting without replacement. We could either compute 10 × 9 × 8 × 7, or notice that this is the same as the permutation
10P4 = 5040.

The probability of no repeated digits is the number of 4 digit PINs with no repeated digits divided by the total number of 4 digit PINs. This probability is 10P4/104 = 5040/ 10000 = 0.504

Combination Formula With Repetition

Sometimes when we find out combinations some objects may be repeated. Where there is repetition allowed formula is changed to (r+n-1)!/r!(n-r)!, where n is the number of things to choose from, and we choose r of them and repetition allowed and order does not matter.

Permutation And Combination Formula List

Some of the formulas for combination are as follows:

  • N! = N(N-1)(N-2)(N-3)……1 ▪ 0!=1!=1
  • nCr = n!/ n−r ! r!
  • nPr= n! /n−r !
  • Selecting r objects out of n is same as selecting (n-r) objects out of n, nCr = nCn-r

Check Out: Probability Formulas


Combination Formula Derivation

Combination formula thus can be derived from multiplication and permutation –

  • No. of ways to select the first object from n distinct objects: n ways
  • No. of ways to select a second object from (n-1) distinct objects:
  • (n-1) ways
  • No. of ways to select a third object from (n-2) distinct objects: (n-2) ways
  • No. of ways to select a fourth object from (n-3) distinct objects: (n-3) ways
  • No. of ways to select rth object from (n-(r-1)) distinct objects: (n-(r-1)) ways

Completing selection r things of the original set of n things creates an ordered sub-set of r elements. 

∴ the number of ways to make a selection of r elements of the original set of elements is (– 1) (– 2) (n-3) . . . (– (– 1)) or (– 1) (– 2) … (– + 1) 

Let us consider the ordered sub-set of r elements and all its permutations. The total number of all permutations of this sub-set is equal to r!  because objects in every combination can be rearranged in r! ways.

Hence, the total number of permutations of different things taken at a time is nCr ×r! On the other hand, it is nPr.

nPr=nCr×r!


Check Out: Quantitative AptitudeVedic Maths


Combination Formula In Python

A python is a coding software. Python is used by software developers. A combination function can also be used in python and can be coded.

Python takes a list and an input r as an input and returns an object list of tuples which contain all possible combination of length r in a list form.

# A Python program to print all 
# combinations of a given length from itertools import combinations
# Get all combinations of [1, 2, 3]
# and length 2 
comb = combinations([1, 2, 3], 2) 
# Print the obtained combinations for i in list(comb): 
print i 
Output
(1, 2)
(1, 3)
(2, 3)

  • Python is used for software development
  • Combination can also be used in python 
  • Coding is done to find combination
  • A list of tuples comes as an output

When To Use Combination Formula?

The combination can be used to solve math problems and also in real life. Lottery Game – In the game of lottery the numbers are selected randomly. Like if someone has to select 4 numbers from the first 14 natural numbers. One has to select the digits without repetition. 

Selecting nominees for student council – In the selection of the nominees of the student council, positions are fixed. Therefore the president can only be selected from the 12th class and other positions are filled from the 11th class, so students are selected by combination.

  • Many uses of permutation and combination
  • Used in daily life also
  • Used for drawing lotteries
  • Selecting balls
  • Finding probability

Check Out: Vedic Maths


Relation Between Permutation And Combination Formula

The combination expression is a permutation divided by x!  Corresponding to each combination of nC , we have r ! permutations, because r r objects in every combination can be rearranged in r! ways.
Hence, the total number of permutations of n different things taken r at a time isnC ⋅r!.On the other hand, it is nP. 

nP=nC r!,0<rn is the relation between permutation and combination. 

Permutations P(n,r) (solutions, examples, videos)

FAQ

What is combination formula?

Ans. A combination is a selection of some or all of several different objects from a collection where the order of selection does not matter or is simply immaterial. The number of selections of r objects from the given n objects is denoted by nCr and is given by nCr = n! / r!(n−r)! .

What is formula for permutations and combinations?

Ans. The Formula for Permutation: P(n,r)=n! (n−r)!. The combination expression is a permutation divided by x!  Corresponding to each combination of nC , we have r ! permutations, because r r objects in every combination can be rearranged in r! ways. Hence, the total number of permutations of n different things taken r at a time isnC ⋅r!.On the other hand, it is nP. 
The formula for combination: The number of selections of r objects from the given n objects is denoted by nCr and is given by nCr = n! / r!(n−r)! .

What are all the possible combinations of 1234?

Ans. The possible combination of 1234 will be 24 and they are: 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2143, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, or 4321.

How do you know when to add or multiply combinations?

Ans. There are ways to identify whether to multiply or add a combination. check out the link to know: https://youtu.be/G_Ngyq2RvAk

How do you calculate the number of possible combinations?

Ans. For calculating the number of possible combinations, the formula is n! / (r! (n — r)!), where n is the total number of possibilities to start and r is the number of selections made. For eg. there are 52 cards, so ‘n’ will be 52.

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Piyush values education and has studied from the top institutes of IIT Roorkee, IIM Bangalore, KTH Sweden and Tsinghua University in China. Post completing his MBA, he has worked with the world’s # 1 consulting firm, The Boston Consulting Group and focused on building sales and marketing verticals for top MNCs and Indian business houses.

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