Friday, 11 November 2016

Download PDF for Permutation and Combination Problems For Govt Exams

Permutation and Combination Formulas 1. Factorial Notation: Let  n  be a positive integer. Then, factorial  n , denoted  n ! is defi... thumbnail 1 summary

Permutation and Combination Formulas

1. Factorial Notation:
Let n be a positive integer. Then, factorial n, denoted n! is defined as:
n! = n(n – 1)(n – 2) … 3.2.1.
Examples:
  1. We define 0! = 1.
  2. 4! = (4 x 3 x 2 x 1) = 24.
  3. 5! = (5 x 4 x 3 x 2 x 1) = 120.
 2. Permutations:
The different arrangements of a given number of things by taking some or all at a time, are called permutations.
Examples:
  1. All permutations (or arrangements) made with the letters abc by taking two at a time are (abbaaccabccb).
  2. All permutations made with the letters abc taking all at a time are:
    ( abcacbbacbcacabcba)
3. Number of Permutations:
Number of all permutations of n things, taken r at a time, is given by:
nPr = n(n – 1)(n – 2) … (n – r + 1) =124
Examples:
  1. 6P2 = (6 x 5) = 30.
  2. 7P3 = (7 x 6 x 5) = 210.
  3. Cor. number of all permutations of n things, taken all at a time = n!.
4. An Important Result:
If there are n subjects of which p1 are alike of one kind; p2 are alike of another kind;p3 are alike of third kind and so on and pr are alike of rth kind, 
such that (p1 + p2 + … pr) = n.
Then, number of permutations of these n objects is = 125
5. Combinations:
Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.
Examples:
  1. Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.Note: AB and BA represent the same selection.
  2. All the combinations formed by abc taking abbcca.
  3. The only combination that can be formed of three letters abc taken all at a time is abc.
  4. Various groups of 2 out of four persons A, B, C, D are:
    AB, AC, AD, BC, BD, CD.
  5. Note that ab ba are two different permutations but they represent the same combination.
6. Number of Combinations:
The number of all combinations of n things, taken r at a time is:
126
Note:
  1. nCn = 1 and nC0 = 1.
  2. nCr = nC(n – r)
Examples:
i.   11C4 = 127
ii.  128


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