Question

Given a seven digit word PRODIGY which is made using 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

Answer 1

Given: A seven digit word PRODIGY which is made using distinct digits 1,2,3,4,5,6,7.

To find:  Sum of digits of S.

Solution:

• Now we have given that there are 5040 words which we can get by arranging the letters of the word PRODIGY.

• Every letter is used once and each letter at each position is used for:

                    5040/7 = 720 times

• Now sum at each place will be:

                    720 (1 + 2 + 3 + 4 + 5 + 6 + 7)

                    720 x 28 = 20160

• Now we can see that the unit place is 0 and 2016 is carried over to  next 20160.

• 20160 + 2016  =  22176

                    So the tens place is 6.

• 2217 is taken now

                    20160 + 2217  =  22377

                    So, 100th place is 7

• 2237 is  taken now

                    20160 + 2237  =  22397

                    1000th place is 7

• 2239 is taken now

                    20160 + 2239  =  22399

                    10000th place is 9

• 2239 is taken now

                    20160 + 2239  =  22399

                    100000th place is 9

• 2239 is taken now

                    20160 + 2239 = 22399

• So the sum comes out to be:

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

                    ( 20160 sum is at each place )

                    S = 20160 ( 1111111 )

                    S = 22399997760‬

• Now the sum of digits is: 2 + 2 + 3 + 9 + 9 + 9 + 7 + 7 + 6 + 0 = 54

Answer:

             So the sum of digits is 54.