Question

given a 7 digit word Prodigy which is made using seven distinct digits 1234 567 there are 5040 words which we can get by arranging the letters of the word Prodigy if the sum of all these numbers then find the sum of digits of s
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Answer 1

Given : 7 digit word Prodigy which is made using seven distinct digits 1234 567 there are 5040 words which we can get by arranging the letters of the word Prodigy

To find : Sum of Digits of S  where S = Sum of all these numbers

Solution:

There are 5040 Words

Each  alphabet is used once in each word

Hence each alphabet is used 5040 times

and each alphabet at each position is used  5040/7  =  720  Times

Sum  at Each place = 720 (1 + 2 + 3 + 4 + 5 + 6 + 7)

= 720 * 7 * 8 /2

= 20160

unit place is  0  

now 2016 is carried over to  next 20160

20160 + 2016  =  22176

Hence tens place = 6

2217 is carried over

20160 + 2217  =  22377

100 s place = 7

2237 is carried over

20160 + 2237  =  22397

1000 s place = 7

2239 is carried over

20160 + 2239  =  22399

10000 s place = 9

2239 is carried over

20160 + 2239  =  22399

100000 s place = 9

2239 is carried over

20160 + 2239  =  22399

Hence sum S = 22,39,99,97,760

or another way to get Sum

S = 20160( 1 + 10 + 100 + 1000 + 10000 + 100000 + 1000000)

as 20160 sum is at each place

= 20160 (  10⁷- 1)/(10 - 1)

= 2240(10000000 - 1)

= 22400000000 - 2240

= 22,39,99,97,760‬

Sum of Digits

=  2 + 2 + 3 + 9 + 9 + 9 + 7 + 7 + 6 + 0

= 54

Sum of digits of S  = 54

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