I am assuming that we cannot repeat a digit.

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First we'll find the number of 4 digit numbers that do not contain a 
5 or a 6 and then

multiply by the number of ways we can insert a 6 in each, which is 5 ways:

For example in the 4 digit number  2371 can have a 6 inserted in 5 ways:

62371, 26371, 23671, 23761, 23716

The number pf 4-digit numbers we can make from {1,2,3,4,7} is

5 ways to choose the 1st digit.
4 ways to choose the 2nd digit.
3 ways to choose the 3rd digit.
2 ways to choose the 4th digit.

5P4 = 5*4*3*2 = 120 ways.

Then we can insert a 6 to make a 5-digit number 5 ways:

Answer 6*360 = 2160 ways.

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-34--

We can choose the place to put the 6 in 3 ways.
That leaves 2 blanks to fill, a leftmost blank and a rightmost blank.
We can choose the digit to put in the leftmost remaining blank in 3 
remaining ways: {1,2,7}
 
We can choose the digit to put in the rightmost remaining blank in 2
ways.

That's 3*3*2 = 18 ways. 

Edwin