Sum of all possible 4 digit numbers formed from the given numbers

Find the sum of all four digit numbers that can be formed using the digits 1,2,3,4,5, each digit appearing at most once.


There are 5P4 = 120 ways of forming 4-digit numbers with each digit appearing at most once. Consider first the one's place. In those 120 numbers, 1,2,3,4, and 5 appear exactly the same number of times, that is, each digit appears 120/5 = 24 times.

For the tens place, same thing: 1,2,3,4,and 5 appear equally the same number of times. Each digit appears 24 times. The argument for the hundreds and thousands places are the same. Thus the sum of all those distinct 120 numbers is
1,000*24*(1+2+3+4+5)+ 100*24*(1+2+3+4+5) + 10*24*(1+2+3+4+5) + 1*24*(1+2+3+4+5) = 1,111*24*(1+2+3+4+5) = 1,111*24*15 = 399,960.

http://www.algebra.com/algebra/homework/Permutations/Permutations.faq.question.389512.html


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