Saturday, June 25, 2016

combinatorics - How many ways can the digits 2,3,4,5,6 be arranged to get a number divisible by 11



How many ways can the digits 2,3,4,5,6 be arranged to get a number divisible by 11



I know that the sum of the permutations of the digits should be divisible by 11. Also, the total number of ways the digits can be arranged is 5!=120.


Answer




Hint. By the divisibility rule by 11 we have to count the arrangements d1,d2,d3,d4,d5 of the digits 2,3,4,5,6 such that d1+d3+d5(d2+d4) is divisible by 11. Notice that
2=2+3+4(5+6)d1+d3+d5(d2+d4)4+5+6(2+3)=10
therefore we should have d1+d3+d5=d2+d4=2+3+4+5+62=10.



In how many ways we can do that?


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