sum of all possible 4 digit numbers formed by digits of 3556

[defeated_soldier]

The sum of all possible 4 digit numbers formed by digits of 3556 ,using each digit only once is :

(1)64427
(2)63327
(3)65297
(4)43521


My solution :

here is a quicker solution method i learnt sometimes back ...

The average number = (max number +min number)/2 =(6553+3556)/2

we can make numbers = 4!/2! (becoz each digit only once )

so , total sum would be (6553+3556)/2 * 4!/2!

However this does not match with the given answer 63327 .

who is wrong ? my book or me ?

if ,i am wrong ...where it is wrong ? whats the correct way to solve it ?

 

[soroban]

Re: sum of all possible 4 digit numbers formed by digits of

Hello, defeated_soldier!

The sum of all possible 4-digit numbers formed by digits of 3556,
using each digit only once, is :

\(\displaystyle (1)\;64,427\;\;\;(2)\;63,327\;\;\;(3)\;65,297\;\;\;(4)\;43,521\)

Your "averaging" method works if all the numbers from 3556 to 6553 are included.

If you write all twelve permutations and "stack" them, we find that
. . each column contains: three 3's, six 5's, and three 6's.

Since \(\displaystyle \,3\times3\,+\,6\times5\,+\,3\times6\:=\:57\), the addition looks like this:

. . . . . . . 5 7
. . . . . .5 7
. . . . 5 7
. . .5 7
. . - - - - - - -
. . 6 3 3 2 7
.

 

[defeated_soldier]

Re: sum of all possible 4 digit numbers formed by digits of

Hello, defeated_soldier!

The sum of all possible 4-digit numbers formed by digits of 3556,
using each digit only once, is :

\(\displaystyle (1)\;64,427\;\;\;(2)\;63,327\;\;\;(3)\;65,297\;\;\;(4)\;43,521\)

Your "averaging" method works if all the numbers from 3556 to 6553 are included.

If you write all twelve permutations and "stack" them, we find that
. . each column contains: three 3's, six 5's, and three 6's.

Since \(\displaystyle \,3\times3\,+\,6\times5\,+\,3\times6\:=\:57\), the addition looks like this:


. . . . . . . 5 7
. . . . . .5 7
. . . . 5 7
. . .5 7
. . - - - - - - -
. . 6 3 3 2 7
.

you told if "all' are included . what do you mean ? i tried to subtract 3556 , 6553

from (6553+3556)/2 * 4!/2! but i could not get the answer out of it ?
can you repair it ?

thanks