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Combination

The number of combinations of 'n' dissimilar things taken 'r' at a time is denoted by nCr or C(n, r). It is defined as, nCr Description: 918.png
 
NOTE :
  • Description: 923.pngDescription: 928.png 
  • Description: 933.png 
  • Description: 938.png 
  • nCx = nCy  x + y = n
  • nCr = nCr+1n+1Cr
  • nCr = Description: 944.pngn–1Cr–1
  • nCr = Description: 949.png (n – r + 1) nCr–1
  • nC1 = nCn–1 = n
Conditional Combinations
  1. Number of combinations of n distinct things taking r Description: 954.png at a time, when k Description: 959.png particular objects always occur =Description: 964.png.
  2. Number of combinations of n distinct objects taking Description: 969.png at a time, when kDescription: 974.png particular objects never occur =Description: 979.png.
  3. Number of selections of r things from n things when p particular things are not together in any selection  nCr – n – pCr – p
  4. Number of selection of r consecutive things out of n things in a row = n – r + 1
  5. Number of selection of r consecutive things out of n things along a circle Description: 984.png 
  6. The number of Combinations of 'n' different things taking some or all at a time Description: 989.png 
  7. The number of ways of dividing 'm + n' things into two groups containing 'm' and 'n' things respectively m + nCm nCn Description: 995.png




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