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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 3…and so on n things at time; hence, the number of ways is nC1 + nC2 + nC3 + … nCn. = 2n–1, 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 ways—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 nth bag in two ways. Using multiplication theorem of counting, the number of ways of dealing with all the bags together is 2 × 2 × 2 ×…n times = 2n 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.
 
Hence, the number of ways of selecting one or more things from n given things is 2n−1.
 
Distributing the given things (m + n) into two groups where one group is having m things and other one n things If we select m things (which can be done in m+nCm ways), then we will be left with 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 m+nCm.
m+nCm Description: 2078.png
 
If we take m = n, then the above expression will denote “Distributing 2m things equally between two distinct groups = 2mCm Description: 2087.png

However, when the groups are identical, then we will be required to divide the above result by 2!.
 
Hence, in that case it becomes Description: 2096.png

(Refer to word formation examples)
 
All the above derivations with their different applications can be seen below in a summarized form.
  1. Fundamental Principle of counting:
    1. 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.
    2. 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.
       
      Thus, if a work is done when exactly one of the number of works is done, then the number of ways of doing this work = sum of the number of ways of doing all the works.
  2. If nCx nCy then either x = y or x + y = n.
  3. Description: 2105.png= 1.2.3…nDescription: 2114.png = 1
  4. a. The number of permutations of n different articles taking r at a time is denoted by nPr and Description: 2123.png
    1. The number of permutations of n different articles taking all at a time = Description: 2132.png.
    2. 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 =Description: 2136.png.
  5. The number of permutations (arrangements) of n different articles taking r at a time when articles can be repeated any number of times = n × n × … r times = nr.
  6. Circular permutations:
    1. The number of circular permutations (arrangements) of n different articles = Description: 2146.png.
    2. 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 Description: 2155.png.
  7. The number or combinations of n different articles taking r at a time is denoted by Description: 2164.png and Description: 2169.png.
  8. The number of selections of r articles (r  n) out of n identical articles is 1.
  9. Total number of selections of zero or more articles from n distinct articles = nC0 + nC1 nC2 + …nCn = 2n.
  10. Total number of selections of zero or more articles from n identical articles = 1 + 1 + 1 +…to (n + 1) terms = n + 1.
  11. The number of ways of distributing n identical articles among r persons when each person may get any number of articles = n+r1Cr–1.
  12. The number of ways of dividing m + n different articles in two groups containing m and n articles respectively (m ≠ n)
     
    Description: 2178.png.
  13. The number of ways of dividing 2m different articles each containing m articles Description: 2187.png.
  14. The number of ways of dividing 3m different articles among three persons and each is getting m articles = Description: 2196.png.
  15. The number of ways of selecting n distinct articles taken r at a time when p particular articles are always included = n-pCr-p.
  16. nCr−1 nCr = n+1Cr
  17. npr = r.n-1pr-1 + n-1Pr




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