How many numbers#1111?

How many 8-digit numbers made from { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } \left\{ 1,2,3,4,5,6,7,8 \right\} which is divisible by 1111 1111 and its digits are distinct?


The answer is 384.

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1 solution

X X
Jun 30, 2018

The number must be a multiple of 9999.If the number is in the form of a b c d e f g h \overline{abcdefgh} ,then a b c d + e f g h = 9999 \overline{abcd}+\overline{efgh}=9999 ,we get a + e = b + f = c + g = d + h = 9 a+e=b+f=c+g=d+h=9 We know 1 + 8 = 2 + 7 = 3 + 6 = 4 + 5 = 9 1+8=2+7=3+6=4+5=9 ,so we'll have to find the possibilities of a b c d \overline{abcd} .

a a has 8 possibilities, b b has 6, c c has 4, d d has 2.So it's 8 × 6 × 4 × 2 = 384 8\times6\times4\times2=384

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