The number of 6 digit numbers that can be formed using the digits 0,1,2,5,7 and 9 which are divisible by 11 and no digit is repeated, is:

If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11 completely.

In order to check whether a number like 2143 is divisible by 11, below is the following procedure.

Group the alternative digits i.e. digits which are in odd places together and digits in even places together. Here 24 and 13 are two groups.
Take the sum of the digits of each group i.e. 2+4=6 and 1+3= 4
Now find the difference of the sums; 6-4=2
If the difference is divisible by 11, then the original number is also divisible by 11. Here 2 is the difference which is not divisible by 11.
Therefore, 2143 is not divisible by 11.

Given digits: 0,1,2,4,5,7,9

Sum of the given digit is 0+1+2+5+7+9 = 24

Let the digits be abcdef

The numbers abcdef is divisible by 11 if

For a number to be divisible by 11 , difference of sum of alternate digits of the number should be of the form

|(a+c+e )-(b+d+f)is a multiple of 11

a+c+e=b+d+f=12

Case 1: {a,c,e}={7,5,0}

{b,d,f}={9,2,1}

So, 2 x 2! x 3! = 24

Case 2: {a,c,e}={9 , 2, 1}

{b,d,f}={7, 5, 0}

So, 3! x 3! = 36

Total = 24 + 36 = 60

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