# Total Number of Combinations

Out of*n*things, the number of ways of selecting one or more things:

where we can select 1 or 2 or 3â€¦and so on

*n*things at time; hence, the number of ways is*C*^{n}_{1}+*C*^{n}_{2}+*C*^{n}_{3}+ â€¦*C*^{n}*. =*_{n}**2***â€“1*^{n}**,**where*n*is the number of things.Above derivation can also be understood in the following manner:

Let there be

*n*bags.The first bag can be dealt in two waysâ€”it is either included or not included. Similarly, the second bag can be dealt in two ways, the third one in two ways and so on, the
Hence, the number of ways of selecting one or more things from
Distributing the given things (m + n) into two groups where one group is having m things and other one n things

*nth*bag in two ways. Using multiplication theorem of counting, the number of ways of dealing with all the bags together is 2 Ã— 2 Ã— 2 Ã—â€¦*n*times = 2*ways. But out of these, there is one combination where we do not include any of the bags. This is not allowed because we have to select at least one thing.*^{n}*n*given things is 2*âˆ’1.*^{n}**If we select***m*things (which can be done in*C*^{m+n}*ways), then we will be left with*_{m}*n*things, i.e., we have two groups of*m*and*n*things respectively. So, the number of ways of dividing (*m*+*n*) things into two groups of*m*and*n*things respectively is equal to*C*^{m+n}*.*_{m}*C*

^{m+n}

_{m}_{ }=

If we take

*m = n*, then the above expression will denote â€œDistributing 2m things equally between two distinct groups**=**^{2m}C*=*_{m}However, when the groups are identical, then we will be required to divide the above result by 2!.

(Refer to word formation examples)

- Fundamental Principle of counting:
*Multiplication rule*If a work is done only when all the number of works are done, then the number of ways of doing that work is equal to the product of the number of ways of doing separate works.*Addition rule*If a work is done only when any one of the number of works is done, then the number of ways of doing that work is equal to the sum of the number of ways of doing separate works.

- If
C^{n}_{x}^{ }=C^{n}_{y}_{ }then either*x*=*y*or*x*+*y*=*n*. - = 1.2.3â€¦
*n*; = 1 - a. The number of permutations of
*n*different articles taking*r*at a time is denoted byP^{n}_{r}_{ }and_{}- The number of permutations of n different articles taking all at a time = .
- The number of permutations of
*n*articles, out of which*p*are alike and are of one type,*q*are alike and are of second type and rest are all different =.

- The number of permutations (arrangements) of
*n*different articles taking*r*at a time when articles can be repeated any number of times =*n Ã— n*Ã— â€¦*r*times =*n*.^{r} - Circular permutations:
- The number of circular permutations (arrangements) of
*n*different articles = . - The number of circular arrangements of
*n*different articles when clockwise and anticlockwise arrangements are not different i.e., when the observation can be made from both the sides .

- The number of circular permutations (arrangements) of
- The number or combinations of
*n*different articles taking*r*at a time is denoted by and . - The number of selections of
*r*articles (*r*â‰¤*n*) out of*n*identical articles is 1. - Total number of selections of zero or more articles from
*n*distinct articles =C^{n}_{0}^{ }+C^{ n}_{1}^{ }+C^{n}_{2}^{ }+ â€¦C^{n}= 2_{n}^{n}^{.} - Total number of selections of zero or more articles from
*n*identical articles = 1 + 1 + 1 +â€¦to (*n*+ 1) terms =*n*+ 1. - The number of ways of distributing
*n*identical articles among*r*persons when each person may get any number of articles =^{n+r}^{â€“}^{1}C_{râ€“}_{1}. - The number of ways of dividing
*m + n*different articles in two groups containing*m*and*n*articles respectively (*m*â‰*n*) - The number of ways of dividing 2
*m*different articles each containing*m*articles . - The number of ways of dividing 3
*m*different articles among three persons and each is getting*m*articles = . - The number of ways of selecting
*n*distinct articles taken*r*at a time when*p*particular articles are always included =^{n}^{-}C^{p}_{r}_{-}_{p}*.* C^{n}_{r}_{âˆ’1}C^{ }+^{n}_{r}=^{n+}^{1}C_{r}_{npr = r.n-1pr-1 + n-1Pr}