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**SUMMARY**

**â‡’ Fundamental principles of counting**

**(a) Multiplication Rule:** If certain thing may be done in â€˜mâ€™ different ways and when

it has been done, a second thing can be done in â€˜n â€˜ different ways then total number

of ways of doing both things simultaneously = m Ã— n.

**(b) Addition Rule:** It there are two alternative jobs which can be done in â€˜mâ€™ ways

and in â€˜nâ€™ ways respectively then either of two jobs can be done in (m + n) ways.

**â‡’ Factorial:** The factorial n, written as n! or n , represents the product of all integers from 1 to n both inclusive. To make the notation meaningful, when n = o, we define o! or o

= 1.

Thus, n! = n (n â€“ 1) (n â€“ 2) â€¦.. â€¦3.2.1

**â‡’** **Permutations:** The ways of arranging or selecting smaller or equal number of persons

or objects from a group of persons or collection of objects with due regard being paid to

the order of arrangement or selection, are called permutations.

The number of permutations of n things chosen r at a time is given by

^{n}P_{r} =n ( n â€“ 1 ) ( n â€“ 2 ) â€¦ ( n â€“ r + 1 )

where the product has exactly r factors.

**â‡’**** Circular Permutations:** (a) n ordinary permutations equal one circular permutation.

Hence there are ^{n}P_{n}/ n ways in which all the n things can be arranged in a circle. This

equals (nâ€“1)!.

(b) the number of necklaces formed with n beads of different colours

- Number of permutations of n distinct objects taken r at a time when a particular object is not taken in any arrangement is
^{nâ€“1}p_{r}. - Number of permutations of r objects out of n distinct objects when a particular object is always included in any arrangement is r.
^{nâ€“1}p_{r-1}.

** â‡’ Combinations:** The number of ways in which smaller or equal number of things are

arranged or selected from a collection of things where the order of selection or

arrangement is not important, are called combinations.

(a) ^{n}C_{r} has a meaning only when r and n are integers 0â‰¤ r â‰¤ n and ^{n}C_{nâ€“r} has a meaning only when 0 â‰¤ n â€“ r â‰¤ n.

Permutations when some of the things are alike, taken all at a time

Permutations when each thing may be repeated once, twice,â€¦upto r times in any

arrangement = n!.

â‡’ The total number of ways in which it is possible to form groups by taking some or all of

n things (2^{n} â€“1).

The total, number of ways in which it is possible to make groups by taking some or all

out of n (=n_{1} + n_{2} + n_{3} +â€¦) things, where n1 things are alike of one kind and so on, is

given by { (n_{1} + 1) ( n_{2} + 1) ( n_{3} + 1)â€¦} â€“1

â‡’ The combinations of selecting r_{1} things from a set having n_{1} objects and r_{2} things from a set having n_{2} objects where combination of r_{1} things, r_{2} things are independent is given

by

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