Given a sever digit word PRODIGY which is made using seven distinct digits 1,2,3,4,5,6,7. There are 5040
words which we can get by arranging the letters of the word PRODIGY. If S be the sum of all these
numbers then find sum of digits of S.
O
45
O 54
O
63
0 72
Answers
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
54 is correct answer
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