How many zeroes will be there at the end of the expression (2!)2! + (4!)4! + (8!)8! = +(9!)9! + (10!)10! +(11!)11!

blank to the power of blank

Asked by Varsha | 5th May, 2015, 09:08: PM

Expert Answer:

T o space f o r m space a space z e r o comma space w e space n e e d space t o space h a v e space f a c t o r s space 2 space a n d space 5. T h e space n u m b e r space o f space z e r o s space a r e space a l w a y s space d e p e n d s space o n space t h e space p a i r space o f space 2 space x space 5 space i n space a n y space n u m b e r A s space t h e r e space i s space n o t space a space sin g l e space 5 space i n space open parentheses 2 factorial close parentheses to the power of 2 factorial end exponent space a n d space open parentheses 4 factorial close parentheses to the power of 4 factorial end exponent comma space open parentheses 2 factorial close parentheses to the power of 2 factorial end exponent space plus space open parentheses 4 factorial close parentheses to the power of 4 factorial end exponent space d o e s space n o t space c o n t a i n space a n y space z e r o e s. N o w space c o n s i d e r space 8 factorial. 8 factorial equals 8 cross times 7 cross times 6 cross times 5 cross times 4 cross times 3 cross times 2 cross times 1 T h u s space i n space 8 factorial comma space m a x i m u m space p o s s i b l e space p a i r space t h a t space c a n space b e space m a d e space i s space 1 space a n d space h e n c e space t h e space n u m b e r space o f space z e r o s space i n 8 factorial space i s space 1. N o w space l e t space u s space f i n d space t h e space n u m b e r space o f space z e r o s space i n space open parentheses 8 factorial close parentheses to the power of 8 factorial end exponent T h u s space t h e space t e r m space open parentheses 8 factorial close parentheses to the power of 8 factorial end exponent space w o u l d space h a v e space 1 cross times 8 factorial equals 8 factorial space z e r o s space i n space t h e space e n d.  N o w space c o n s i d e r space 9 factorial. 9 factorial equals 9 cross times 8 cross times 7 cross times 6 cross times 5 cross times 4 cross times 3 cross times 2 cross times 1 T h u s space i n space 9 factorial comma space m a x i m u m space p o s s i b l e space p a i r space t h a t space c a n space b e space m a d e space i s space 1 space a n d space h e n c e space t h e space n u m b e r space o f space z e r o s space i n 9 factorial space i s space 1. N o w space l e t space u s space f i n d space t h e space n u m b e r space o f space z e r o s space i n space open parentheses 9 factorial close parentheses to the power of 9 factorial end exponent T h u s space t h e space t e r m space open parentheses 9 factorial close parentheses to the power of 9 factorial end exponent space w o u l d space h a v e space 1 cross times 9 factorial equals 9 factorial space space z e r o s space i n space t h e space e n d.  N o w space c o n s i d e r space 10 factorial. 10 factorial equals 10 cross times 9 cross times 8 cross times 7 cross times 6 cross times 5 cross times 4 cross times 3 cross times 2 cross times 1 T h u s space i n space 10 factorial comma space m a x i m u m space p o s s i b l e space p a i r space t h a t space c a n space b e space m a d e space a r e space 2 space a n d space h e n c e space t h e space n u m b e r space o f space z e r o s space i n 10 factorial space i s space 2. N o w space l e t space u s space f i n d space t h e space n u m b e r space o f space z e r o s space i n space open parentheses 10 factorial close parentheses to the power of 10 factorial end exponent T h u s space t h e space t e r m space open parentheses 10 factorial close parentheses to the power of 10 factorial end exponent space w o u l d space h a v e space 2 cross times 10 factorial space z e r o s space i n space t h e space e n d.  N o w space c o n s i d e r space 11 factorial. 11 factorial equals 11 cross times 10 cross times 9 cross times 8 cross times 7 cross times 6 cross times 5 cross times 4 cross times 3 cross times 2 cross times 1 T h u s space i n space 11 factorial comma space m a x i m u m space p o s s i b l e space p a i r space t h a t space c a n space b e space m a d e space a r e space 2 space a n d space h e n c e space t h e space n u m b e r space o f space z e r o s space i n 10 factorial space i s space 2. N o w space l e t space u s space f i n d space t h e space n u m b e r space o f space z e r o s space i n space open parentheses 11 factorial close parentheses to the power of 11 factorial end exponent T h u s space t h e space t e r m space space open parentheses 11 factorial close parentheses to the power of 11 factorial end exponent space w o u l d space h a v e space 2 cross times 11 factorial space z e r o s space i n space t h e space e n d.  T h e r e f o r e space t h e space t o t a l space s u m space o f space t h e space e x p r e s s i o n space open parentheses 2 factorial close parentheses to the power of 2 factorial end exponent plus space open parentheses 4 factorial close parentheses to the power of 4 factorial end exponent plus open parentheses 8 factorial close parentheses to the power of 8 factorial end exponent plus open parentheses 9 factorial close parentheses to the power of 9 factorial end exponent plus space open parentheses 10 factorial close parentheses to the power of 10 factorial end exponent plus open parentheses 11 factorial close parentheses to the power of 11 factorial end exponent equals 4 plus 24 to the power of 24 plus open parentheses 8 factorial close parentheses to the power of 8 factorial end exponent plus open parentheses 9 factorial close parentheses to the power of 9 factorial end exponent plus space open parentheses 10 factorial close parentheses to the power of 10 factorial end exponent plus open parentheses 11 factorial close parentheses to the power of 11 factorial end exponent  U n i t space d i g i t space o f space 24 to the power of 24 space i s space 6 space u sin g space c y c l i c i t y space o f space p o w e r s space o f space 24. U n i t space d i g i t space o f space s u m space o f space 4 space a n d space 24 to the power of 24 space i s space 10. H e n c e space w e space w i l l space g e t space o n e space z e r o space i n space t h e space e n d space w i t h space a space c a r r y space o f space 1 space t o space t h e space t e n apostrophe s space p l a c e. S o comma space t h e space n u m b e r space o f space t r a i l i n g space z e r o space o f space t h e space g i v e n space e x p r e s s s i o n space i s space 1.

Answered by Vimala Ramamurthy | 6th May, 2015, 10:47: AM

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