Q6.
(a). What is permutation? Explain circular permutation with example. (5 Marks)
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Permutation: Suppose we have 15 Books that we want to arrange on a sheaf. How many ways are these of doing it? Using the multiplication principle you would say
15*14*13*......*2*1=15!
Each of these arrangements of the books.
Definition: An arrangement of a set of n objects in a given order is called a permutation of the objects
An ordered arrangement of the n objects, taking r at a time(r<=n) is called a permutation of the n objects taking r at a time.
The total number of permutation is P(n,r)=n!/(n-r)!
Circular permutation:
Consider an arrangement of 4 objects, a, b, c and d. We observe the objects in the clockwise direction. On the Circumference these is no preferred origin and hence the permutation abcd, bcda, cdab, dabc will look exactly alike. So each linear permutation when treated as a circular permutation is repeated 4 times.
Thus, the number of circular permutation of n things taken all at a time , is (n-1)!
Example: In how many distinct may is it possible to seat eight persons at a round table?
Solution: Clearly we need the number of circular permutation of 8 things. Hence the number is 7! = 5040.
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(b). Find inverse of the following function:
f(x)=(x2+5)/(x-3) where x is not equal 3
(2 Marks)
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Step 1: Replace f(x) by y in the equation describing the function.
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(c). What is a function? Explain the uses of the functions. (3 Marks)
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Function:
Because functions are used in so many area of mathematics and in so many different ways no single definition of function has been universally adopted some a function is a way to assign to each element of the given set exactly one element of a given set. A function is a special kind of relation. R={(a,b) belong AXB| b is roll number of a} This is a relation between A and B It is a 'special' relation, 'special' because to each a belong A there exits ! b such that aRb. we call such a relation a function A to B .
Definition:
1. A function from a non empty set A to a non -empty set B is a subset R of AXB such that for each a belong A there exits a unique b belong B such that (a,b) belong R.
2. Let A and B be non empty sets. A function f from A to B is a roll that assign to each element x in A exactly one element y is B.
f:A --> B (f is a function from A to B).
Example: If A={1,2,3,4} , B={1,8,27,64,125} and the rule of f assign to each number in A its cube. then f is a function of f is A, its co domain is B and its range is {1,8,27,64,125}.
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