sum of all

[sanju09]

Find the sum of all different 5-digit numbers that can be formed by using the digits 2, 3, 4, 5, and 6, where no digit is repeating.

[adelx]

I think the answer is: 6! * 17 = 720 * 17 = 12240

[sanju09]

I think the answer is: 6! * 17 = 720 * 17 = 12240

please elaborate it a bit

[kapsii]

We have 5 digits and 5 places to fill them, which makes it easier, each digit can occupy either the units place, tens place, hundreds place, thousands place or ten thousands place.
Thus, in all combinations possible, sum of the digit 2 in all the possible places with proper weights would be:
2*10^4 + 2*10^3 + 2*10^2 + 2*10^1 + 2*10^0
Similarly for other digits, hence, together all our digits would contribute to:
(2+3+4+5+6) * 10^4
(2+3+4+5+6) * 10^3
(2+3+4+5+6) * 10^2
(2+3+4+5+6) * 10^1
(2+3+4+5+6) * 10^0

Or 20000 + 2000 + 200 + 20 + 2 = 22222

[sanju09]

kapsii! You are near there, see your calculations only...

[sanju09]

Actually each of the numbers would be in any place 4! times. Hence their contribution when in the ten thousand's place is
(2 + 3 + 4 + 5 + 6) (10000) * 4!
Similarly, when in thousand's place, they contribute
(2 + 3 + 4 + 5 + 6) (1000) * 4!
For 100s, 10s, and 1s places, the contributions are
(2 + 3 + 4 + 5 + 6) (100) * 4!
(2 + 3 + 4 + 5 + 6) (10) * 4!
(2 + 3 + 4 + 5 + 6) (1) * 4!
respectively.
Hence, the total contribution to the sum is

[spoiler](2 + 3 + 4 + 5 + 6) (11111) * 4![/spoiler]

[ssmiles08]

Hi newbie here...I approached the problem the following way... and I got a different answer. Can someone tell me what I might be doing wrong here?

5 numbers can be formed by using 2, 3, 4, 5, 6 = that makes it 120 possible combinations of 5 digit numbers.

Then I took the average of the smallest number and the largest number possible : (65432 + 23456)/2 = 44444


avg = (sum)/(#)

so 44444 = (sum)/(120)

sum = 5333280

[sanju09]

Hi newbie here...I approached the problem the following way... and I got a different answer. Can someone tell me what I might be doing wrong here?

5 numbers can be formed by using 2, 3, 4, 5, 6 = that makes it 120 possible combinations of 5 digit numbers.

Then I took the average of the smallest number and the largest number possible : (65432 + 23456)/2 = 44444


avg = (sum)/(#)

so 44444 = (sum)/(120)

sum = 5333280

Average (arithmetic mean) of the first and last entries will be the average of the entire data-list only if the data is forming an AP; present is not that case perhaps!

[gabriel]

Hi newbie here...I approached the problem the following way... and I got a different answer. Can someone tell me what I might be doing wrong here?

5 numbers can be formed by using 2, 3, 4, 5, 6 = that makes it 120 possible combinations of 5 digit numbers.

Then I took the average of the smallest number and the largest number possible : (65432 + 23456)/2 = 44444


avg = (sum)/(#)

so 44444 = (sum)/(120)

sum = 5333280

Like Sanju has explained the formula mean*number of elements = sum of all the elements is true only if the numbers are consecutive or in an AP (consecutive numbers are also in an AP). In the question since we are only given the 5 numbers this clearly is not the case.

[ssmiles08]

aha gotchya! thanks