Combinatorics #5

first of all, there would be only 12 numbers which are divisible by 11 and can be formed from the digits 1,3,4,7,9 they are

73491,79431,73194,79134 43791,49731,43197,49137 13794,19734,13497,19437

their sum= 557568 next find sum of all the numbers which can be formed from these digits (this is quite simple..)

unit's digit of this sum would be units digit of (1+3+4+7+9)*24 = 6

ten's digit would be the units digit of (1+3+4+7+9)*24 + 57(carry of previous ) and so on ...

[i've multiplied by 24 as each digit would come 4! times] so we get the total sum = 6399936

thus, the required answer = 6399936 - 557568 = 5842368

24x(11111+33333+44444+77777+99999) - 6x(3030+9090) - 4x(10101+40404+70707)

120 11111 24/5 - 12 (10101 4 + 1010*6)

That's essentially the same solution as Xin Liang & Shiv Gupta, arrived at by subtracting the sum of multiples of 11 from the grand total. I got the grand total as 120 permutations of 5 digit. By symmetry, each occurs an equal number of times in each digit position, so the average digit value is 24/5, multiplied by 11111 to apply that average value to each digit position.

Second, a number is divisible by 11 iff the sum of even digits positions differs from sum of odd digit positions by a multiple of 11. Since the digits sum to 24, those (even,odd) sums must be (1,23), (12,12) or (23,1). Only (12,12) is possible, and the only way to make 12 from two of those digits. The even digits must be 9 and 3, with (1,4,7) in the other three spots. There are 2!*3!=12 such numbers.

The digits in the odd positions average (1+4+7)/3=4 for a total weight of 12 10101 4 in the divisible-by-11 sum. The digits in the even positron average (9+3)/2-5 for a total weight of 12 1010 6. The final expression above follows immediately.

[This is not a complete solution.]

There is an easy approach to this problem that doesn't require listing out every 5-digit number and then testing them for divisibility.

Hint: Ignore the condition on multiples of 11. What would be the sum of all the 5-digit integers whose digits are 1, 3, 4, 7, 9 with each of these digits appearing exactly once?