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    Question

    If 9P5 + 5 9P4 = scriptbase P subscript r end scriptbase presuperscript 10, then r =

    1. 4    
    2. 5    
    3. 9    
    4. 10    

    Hint:

    Here we need to find the value of the given variable. Here we will first apply the formula of the permutation to simplify the equation. Then we will apply the mathematical operations that will be required to simplify the given expression further. After simplifying the terms, we will get the value of the required variable.
    Formula used:
    scriptbase P subscript r space equals space fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial end fraction end scriptbase presuperscript n comma space r space i s space t h e space n u m b e r space o f space o b j e c t s space s e l e c t e d space a n d space n space t o t a l space n u m b e r space o f space e l e m e n t s space i n space t h e space s e t.

    The correct answer is: 5

      Detailed Solution
       Here we need to find the value of the given variable.
      The given expression is :
      scriptbase P subscript 5 space plus space 5. end scriptbase presuperscript 9 scriptbase P subscript 4 space equals space scriptbase P subscript r end scriptbase presuperscript 10 end scriptbase presuperscript 9

        N o w comma space w e space w i l l space u s e space t h i s space f o r m u l a space o f space p e r m u t a t i o n space scriptbase P subscript r space equals space fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial end fraction end scriptbase presuperscript n comma space i n space t h e space a b o v e space e q u a t i o n T h e r e f o r e comma space w e space g e t space rightwards double arrow fraction numerator 9 factorial over denominator left parenthesis 9 minus 5 right parenthesis factorial end fraction space plus space 5. space fraction numerator 9 factorial over denominator left parenthesis 9 minus 4 right parenthesis factorial end fraction space equals space fraction numerator 10 factorial over denominator left parenthesis 10 minus r right parenthesis factorial end fraction space space space space rightwards double arrow fraction numerator 9 factorial over denominator left parenthesis 4 right parenthesis factorial end fraction space plus space 5. space fraction numerator 9 factorial over denominator left parenthesis 5 right parenthesis factorial end fraction space equals space fraction numerator 10 factorial over denominator left parenthesis 10 minus r right parenthesis factorial end fraction D i v i d i n g space b o t h space s i d e s space b y space 9 factorial space w e space g e t rightwards double arrow fraction numerator 1 over denominator left parenthesis 4 right parenthesis factorial end fraction space plus space 5. space fraction numerator 1 over denominator left parenthesis 5 right parenthesis factorial end fraction space equals space fraction numerator 10 over denominator left parenthesis 9 minus r right parenthesis factorial end fraction............. space space space space space space space space space open parentheses fraction numerator 10 factorial over denominator 9 factorial end fraction space equals space fraction numerator 10 space cross times 9 factorial over denominator 9 factorial end fraction space equals space 10 close parentheses rightwards double arrow fraction numerator 1 over denominator left parenthesis 4 right parenthesis factorial end fraction space plus space 5. space fraction numerator 1 over denominator 5 space cross times 4 factorial end fraction space equals space fraction numerator 10 over denominator left parenthesis 9 minus r right parenthesis factorial end fraction rightwards double arrow fraction numerator 1 over denominator left parenthesis 4 right parenthesis factorial end fraction space plus space space fraction numerator 1 over denominator 4 factorial end fraction space equals space fraction numerator 10 over denominator left parenthesis 9 minus r right parenthesis factorial end fraction rightwards double arrow fraction numerator 2 over denominator 4 factorial end fraction space equals space fraction numerator 10 over denominator left parenthesis 9 minus r right parenthesis factorial end fraction D i v i d i n g space b y space 2 space o n space b o t h space s i d e s rightwards double arrow fraction numerator 1 over denominator 4 factorial end fraction space equals space fraction numerator 5 over denominator left parenthesis 9 minus r right parenthesis factorial end fraction O n space c r o s s space m u l t i p l y i n g rightwards double arrow left parenthesis 9 minus r right parenthesis factorial space equals space 5 space cross times 4 factorial rightwards double arrow left parenthesis 9 minus r right parenthesis factorial space equals space 5 factorial O n space e q u a t i n g space t h e space t w o space f a c t o r i a l s comma space w e space g e t rightwards double arrow 9 space minus space r space equals space 5 rightwards double arrow r space equals space 4

      H e r e space w e space h a v e space o b t a i n e d space t h e space v a l u e space o f space t h e space v a r i a b l e space u s e d space i n space t h e space e x p r e s s i o n. W e space h a v e space a l s o space u s e d 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 space h e r e. space H e r e comma space scriptbase P subscript r space m e a n s space t h e space n u m b e r space o f space w a y s space t o space c h o o s e space r space space d i f f e r e n t space t h i n g s space f r o m space t h e space s e t space o f space n space space t h i n g s space w i t h o u t space r e p l a c e m e n t. space end scriptbase presuperscript n H e r e space w e space n e e d space t o space k n o w space t h e space m e t h o d space o f space f i n d i n g space t h e space f a c t o r i a l space o f space a n y space n u m b e r. space F a c t o r i a l space i s space d e f i n e d space a s space t h e space m u l t i p l i c a t i o n space o f space a l l space t h e space w h o l e space n u m b e r s space f r o m space t h e space g i v e n space n u m b e r space d o w n space t o space t h e space n u m b e r space 1.

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