Maths-
General
Easy

Question

The sum of all possible numbers greater than 10,000 formed by using {1,3,5,7,9} is

  1. 6666600    
  2. 932460    
  3. 6660000    
  4. 1968    

hintHint:

First of all find the total number of numbers satisfying the given conditions. Now fix 4 out of 5 digits each time and divide total numbers into 5 categories and then consider the place value of each digit of 5 categories to find the total sum of numbers.

The correct answer is: 6666600


     Detailed Solution
    In this, we have to find the sum of all numbers greater than 10000 formed by using digits 1, 3, 5, 7, 9, no digit being repeated in any number. We know that any number greater than 10000 would be of 5 digits or more than 5 digits. Also, since we have only 1, 3, 5, 7, 9, and repetition is not allowed, we can make only 5 digit numbers and any number formed by these five digits would be greater than 10000.

    Any of the five digits can occupy ten thousand places. Since repetition is not allowed any of the remaining 4 digits can occupy a thousand’s place. Similarly, any of the remaining 3 digits can occupy a hundred’s place.
    Now, ten’s and unit’s place can be occupied by any of the remaining two digits and last remaining digit respectively.
    5 over 10 to the power of 4 cross times 4 over 10 cubed cross times 3 over 10 squared cross times 2 over 10 to the power of 1 cross times 1 over 10 to the power of 0 equals 120

    T h e r e f o r e comma space t h e space t o t a l space n u m b e r s space f o r m e d space equals space 120.
space N o w comma space t h e space n u m b e r space o f space w a y s space f o r space a space f i x e d space d i g i t space a t space t h e space p l a c e space o f space 10 to the power of 4 space space i s

    blank over 10 to the power of 4 cross times 4 over 10 cubed cross times 3 over 10 squared cross times 2 over 10 to the power of 1 cross times 1 over 10 to the power of 0 equals 24 space w a y s

T h e space n u m b e r space o f space w a y s space f o r space a space f i x e d space d i g i t space a t space t h e space p l a c e space o f space 10 cubed space space i s

4 over 10 to the power of 4 cross times 3 over 10 cubed cross times blank over 10 squared cross times 2 over 10 to the power of 1 cross times 1 over 10 to the power of 0 equals 24 space w a y s

T h e space n u m b e r space o f space w a y s space f o r space a space f i x e d space d i g i t space a t space t h e space p l a c e space o f space 10 squared space space i s

4 over 10 to the power of 4 cross times blank over 10 cubed cross times 3 over 10 squared cross times 2 over 10 to the power of 1 cross times 1 over 10 to the power of 0 equals 24 space w a y s

T h e space n u m b e r space o f space w a y s space f o r space a space f i x e d space d i g i t space a t space t h e space p l a c e space o f space 10 to the power of 1 space space i s

4 over 10 to the power of 4 cross times 3 over 10 cubed cross times 2 over 10 squared cross times blank over 10 to the power of 1 cross times 1 over 10 to the power of 0 equals 24 space w a y s

T h e space n u m b e r space o f space w a y s space f o r space a space f i x e d space d i g i t space a t space t h e space p l a c e space o f space 10 to the power of 0 space space i s

4 over 10 to the power of 4 cross times 3 over 10 cubed cross times 2 over 10 squared cross times 1 over 10 to the power of 1 cross times blank over 10 to the power of 0 equals 24 space w a y s

A s space s h o w n comma space e v e r y space f i x e d space d i g i t space p l a c e d space a t space t h e space p l a c e space o f space 10 to the power of 4 space comma space w e space c a n space f i l l space o t h e r space d i g i t s space a t space o t h e r space p l a c e s space i n space 24 space w a y s space w i t h o u t space r e p e t i t i o n.

H e n c e comma space t h e space s u m space o f space t h e space p l a c e space v a l u e s space o f space 10 to the power of 4 space equals space left parenthesis 1 plus 3 plus 5 plus 7 plus 9 right parenthesis cross times 24 cross times 10 to the power of 4 space.... left parenthesis i right parenthesis

S i m i l a r l y comma space t h e space s u m space o f space t h e space p l a c e space v a l u e s space o f space 10 to the power of 3 space end exponent equals space left parenthesis 1 plus 3 plus 5 plus 7 plus 9 right parenthesis cross times 24 cross times 10 cubed space.... left parenthesis i i right parenthesis

S i m i l a r l y comma space t h e space s u m space o f space t h e space p l a c e space v a l u e s space o f space 10 squared space equals space left parenthesis 1 plus 3 plus 5 plus 7 plus 9 right parenthesis cross times 24 cross times 10 squared space.... left parenthesis i i i right parenthesis

S i m i l a r l y comma space t h e space s u m space o f space t h e space p l a c e space v a l u e s space o f space 10 to the power of 1 space equals space left parenthesis 1 plus 3 plus 5 plus 7 plus 9 right parenthesis cross times 24 cross times 10 to the power of 1 space.... left parenthesis i v right parenthesis right square bracket

S i m i l a r l y comma space t h e space s u m space o f space t h e space p l a c e space v a l u e s space o f space 10 to the power of 0 space end exponent equals space left parenthesis 1 plus 3 plus 5 plus 7 plus 9 right parenthesis cross times 24 cross times 10 to the power of 0 space.... left parenthesis v right parenthesis

S u m space o f space a l l space n u m b e r s space equals space left parenthesis i right parenthesis space plus space left parenthesis i i right parenthesis space plus space left parenthesis i i i right parenthesis space plus space left parenthesis i v right parenthesis space plus space left parenthesis v right parenthesis

equals space 25 cross times 24 cross times 10 to the power of 4 plus 25 cross times 24 cross times 10 cubed plus 25 cross times 24 cross times 10 squared plus 25 cross times 24 cross times 10 to the power of 1 plus 25 cross times 24 cross times 10 to the power of 0

equals space left parenthesis 25 cross times 24 right parenthesis left square bracket space 10 to the power of 4 plus 10 cubed plus 10 squared plus 10 to the power of 1 plus 10 to the power of 0 right square bracket

equals space left parenthesis space 25 cross times 24 right parenthesis cross times left parenthesis 11111 right parenthesis

equals space 6666600


     Therefore, we get the sum of total numbers greater than 10000 formed by using digits 1, 3, 5, 7, 9 without repetition is 6666600.


     

    Here, some students randomly select the numbers and try to add them which is very lengthy as there are 120 numbers and can lead to mistakes. Also, some students do not consider the place value but in questions such as addition, subtraction, etc. of numbers, place value method is most promising.
     
     
     
     
     

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