IMPORTANT FACTS AND FORMULAE
Factorial Notation: Let n be a positive integer. Then, factorial n is denoted by n! and is defined as:
n! = n(n-1)(n-2)........3.2.1.
Examples: (i) 5! = (5x 4 x 3 x 2 x 1) = 120; (ii) 4! = (4x3x2x1) = 24 etc.
We define, 0! = 1.
Permutations: The different arrangements of a given number of things by taking some or all at a time, are called permutations.
Ex. 1. All permutations (or arrangements) made with the letters a, b, c by taking two at a time are:
(ab, ba, ac, bc, cb).
Ex. 2. All permutations made with the letters a,b,c, taking all at a time are:
(abc, acb, bca, cab, cba).
Number of Permutations: Number of all permutations of n things, taken r at a time, given by:
nPr = n(n-1)(n-2).....(n-r+1) = n!/(n-r)!
Examples: (i) 6p2 = (6x5) = 30. (ii) 7p3 = (7x6x5) = 210.
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.
Ex. 1. Let we want to select two boys out of three boys A, B, C. Then, the possible selections are AB, BC and CA.
Note :: AB and BA represent the same selection.
Ex. 2. All combinations formed by a, b, c, taking two at a time are ab, bc, ca.
Ex. 3. The only combination that can be formed of three letters say a, b, c taken all at a time is :: abc.
Ex. 4. Various groups of two out of four persons say A, B, C, D are: AB, AC, AD, BC, BD, CD.
Ex. 5. Note :: ab and ba are two different permutations but they represent same combination.
Number of Combinations: The number of all the combination of n things, taken r at a time is:
nCr = n! / (r!)(n-r)! = n(n-1)(n-2).....to r factors / r!
Note that: ncr = 1 and nc0 = 1.
An Important Result: ncr = nc(n-r).
Example: (i) 11c4 = (11x10x9x8)/(4x3x2x1) = 330.
(ii) 16c13 = 16c(16-13) = 16x15x14/3! = 16x15x14/3x2x1 = 560.
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In Aptitude, Concepts, Permutation And Combination
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