Maths-

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Question

The number of different signals that can be made by 5 flags from 8 flags of different colours is :

6720
$^{8} C_{5}$
$8^{5}$
$5^{8}$

hint Hint:

$W e space a r e space p r o v i d e d space w i t h space 8 space space f l a g s space o f space d i f f e r e n t space c o l o u r s space a n d space 5 space space f l a g s. W e space n e e d space t o space f i n d space t h e space t o t a l space n u m b e r space o f space s i g n a l s space t h a t space c a n space b e space m a d e space b y space t h e space n u m b e r space o f space a r r a n g e m e n t s space o f space 8 space space f l a g s space b y space t a k i n g space 5 space space f l a g s space a t space a space t i m e. A r r a n g i n g space t h i n g s space i s space c a l l e d space p e r m u t a t i o n comma space t h u s space w e space c a n space c l e a r l y space s a y space t h a t space t h e space g i v e n space q u e s t i o n space i s space b a s e d space o n space t h e space t o p i c space “ p e r m u t a t i o n ”. space H e r e space w e space h a v e space t o space a r r a n g e space 8 space space f l a g s space t a k e n space 5 space space f l a g s space a t space a space t i m e space w h i c h space i s space e q u i v a l e n t space t o space f i l l i n g space 8 space space f l a g s space p l a c e d space o u t space o f space 5 space space f l a g s. space space F o r space t h a t space w e space h a v e space t h e space f o r m u l a space o f space p e r m u t a t i o n colon space scriptbase P subscript r end scriptbase presuperscript n space.$

The correct answer is: 6720

$D e t a i l e d space S o l u t i o n W e space h a v e space b e e n space t o l d space t h a t space t h e r e space a r e space 8 space space f l a g s space o f space d i f f e r e n t space c o l o u r s space b y space t a k i n g space 5 space space f l a g s space a t space a space t i m e. T h u s space t h e space t o t a l space n u m b e r space o f space f l a g s space n equals 8 space space a n d space f l a g s space t o space b e space t a k e n space a t space a space t i m e space r equals 5 space. space B y space a p p l y i n g space t h e space p e r m u t a t i o n space f o r m u l a space equals scriptbase P subscript r end scriptbase presuperscript n space equals scriptbase P subscript 5 end scriptbase presuperscript 8$

$B y space a p p l y i n g space i n space p e r m u t a t i o n comma equals fraction numerator space 8 factorial over denominator left parenthesis 8 minus 5 right parenthesis factorial space space end fraction S u b t r a c t i n g space t h e space v a l u e s space i n space d e n o m i n a t o r comma equals space space fraction numerator 8 factorial over denominator 3 factorial space space end fraction E x p a n d i n g space t h e space f a c t o r i a l comma equals space fraction numerator 8 cross times 7 cross times 6 cross times 5 cross times 4 cross times 3 cross times 2 cross times 1 over denominator 3 cross times 2 cross times 1 space space end fraction space equals space 6720 T h u s comma space space t h e space r e q u i r e d space n u m b e r space o f space s i g n a l s space c a n space b e space m a d e space b y space 5 space space f l a g s space f r o m space 8 space space f l a g s space o f space d i f f e r e n t space c o l o u r s space i s space 6720 space.$

Each of the different arrangements which can be made by taking some or all of a number of things at a time is called permutation.
$space T h e space n u m b e r space o f space p e r m u t a t i o n space w i t h o u t space a n y space r e p e t i t i o n space s t a t e s space t h a t space a r r a n g i n g space n space space o b j e c t s space t a k e n space r space space a t space a space t i m e space i s space e q u i v a l e n t space t o space f i l l i n g space r space space p l a c e s space o u t space o f space n space space t h i n g s space equals space n left parenthesis n minus 1 right parenthesis left parenthesis n minus 2 right parenthesis... left parenthesis n minus top enclose r minus 1 end enclose right parenthesis space space space w a y s equals fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial end fraction space equals space scriptbase P subscript r end scriptbase presuperscript n$

$W e space c a n space a l s o space s a y space t h a t comma space space scriptbase P subscript r end scriptbase presuperscript n equals space n left parenthesis n minus 1 right parenthesis left parenthesis n minus 2 right parenthesis... left parenthesis n minus r plus 1 right parenthesis space space comma space t h u s comma scriptbase P subscript 5 end scriptbase presuperscript 8 equals space 8 cross times 7 cross times 6 cross times 5 cross times 4 space equals space 6720 space.$

Thus we can say that the permutation concerns both the selection and the arrangement of the selected things in all possible ways.

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The number of different signals that can be made by 5 flags from 8 flags of different colours is :

6720

The correct answer is: 6720

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6720
$^{8} C_{5}$
$8^{5}$
$5^{8}$

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The partial fractions of $fraction numerator 1 over denominator x to the power of 3 end exponent left parenthesis x plus 2 right parenthesis end fraction$ are

The partial fractions of $fraction numerator 1 over denominator x to the power of 3 end exponent left parenthesis x plus 2 right parenthesis end fraction$ are

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Reason : f(x) = sgn x is always a positive function.

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Reason : f(x) = sgn x is always a positive function.