The shortest side of a triangle must be longer than the difference
between the other two sides.

Therefore the 1 inch stick cannot be used at all because it is not
longer than the difference between any of the other sticks, for
even if they differ by only 1, 1 is not longer than 1.

2 can be the shortest side if the other two sides differ by only 1.
So we can have a triangle with (2,3,4), (2,4,5) or (2,5,6).

3 can be the shortest side if the other two sides differ by either 
1 or 2.  So we can have a triangle with (3,4,5), (3,4,6), or (3,5,6).

4 can only be the shortest side with (4,5,6)

So 7 triangles can be made with the 6 sticks 1,2,3,4,5,6.

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Yes there are two formulas. One is for when there are an
even numbers of sticks and the other is for when there 
are an odd number of sticks. 

If you have sticks of lengths 1,2,3,...,n inches

The formula for when n is even is  n(n-2)(2n-5)/24

The formula for when n is odd is (n-1)(n-3)(2n-1)/24  

The case above is when there are 6 sticks. Since 6 is even,
we use the formula

n(n-2)(2n-5)/24

6(6-2)(2·6-5)/24 = 6(4)(12-5)/24 = 6(4)(7)/24 = 7

Edwin