Quants Permutation & Combination

permutation and combination
  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.
  • 5! = (5 x 4 x 3 x 2 x 1) = 120.

 

  1. 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 abcby taking two at a time are (abbaaccabccb).
  2. All permutations made with the letters abctaking all at a time are:
    ( abcacbbacbcacabcba)
  1. 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) = n!
(n – r)!

Examples:

 

  1. 6P2= (6 x 5) = 30.
  2. 7P3= (7 x 6 x 5) = 210.
  • number of all permutations of nthings, taken all at a time = n!.

 

  1. 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 = n!
(p1!).(p2)!…..(pr!)
  1. 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.

  1. All the combinations formed by abctaking abbcca.

 

  1. The only combination that can be formed of three letters abctaken all at a time is abc.
  2. Various groups of 2 out of four persons A, B, C, D are:

AB, AC, AD, BC, BD, CD.

  1. Note that abba are two different permutations but they represent the same combination.

 

  • Number of Combinations:

The number of all combinations of n things, taken r at a time is:

nCr = n! = n(n – 1)(n – 2) … to r factors .
(r!)(n – r)! r!

Note:

  • . nCn= 1 and nC0 = 1.
  1. nCrnC(n – r)

Examples:

i.   11C4 = (11 x 10 x 9 x 8) = 330.
(4 x 3 x 2 x 1)

 

ii.   16C13 = 16C(16 – 13) = 16C3 = 16 x 15 x 14 = 16 x 15 x 14 = 560.
3! 3 x 2 x 1

 

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