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30. PERMUTATIONS AND COMBINATIONS
IMPORTANT FACTS AND FORMULAE
Factorial Notation:
Let n be a positive integer.
Then, factorial n, denoted by n!
is
defined as:
n! = n(n1)(n2).
.......
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:
n
P
r
= n(n1)(n2).
....(nr+1) = n!/(nr)!
Examples:
(i)
6
p
2
= (6x5) = 30. (ii)
7
p
3
= (7x6x5) = 210.
Cor. Number of all permutations of n things, taken all at a time = n!
An Important Result:
If there are n objects of
which p
1
are alike of
one kind; p
2
are
alike of another kind; p
3
are alike of third kind and
so on and p
r
are alike of rth kind, such
that (p
1
+p
2
+.......
p
r
) = n.
Then, number of permutations of these n objects is:
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This note was uploaded on 08/13/2011 for the course FSD 011 taught by Professor Vinh during the Spring '11 term at Beacon FL.
 Spring '11
 vinh

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