You have $5$ tones of orange, $7$ of green and $4$ of purple. You want to choose two colours of different tones. How many choices do you have?
Orange and green $$5\cdot7$$
Orange and purple
$$5 \cdot 4$$
Green and purple
$$7\cdot 4$$
Is the answer
$$(5\cdot7)+(5\cdot4)+(7\cdot4)$$ ?
Yeah as you can see I tend to have problems deciding whether to add or multiply.
There are two approaches I can think of: The first one is looking at the three different color combinations (orange,green), (orange,purple),(green,purple). Evaluating each of these is simple. they are $5\cdot7,5\cdot4$ and $7\cdot4$ respectively, so the answer is the sum of these, as you say, which is 83 ( I think this is what you did and it is correct).
A second approach consists in counting all the combinations of hues,( If it was possible to put two of the same color together). There are $\frac{16\cdot15}{2}=8\cdot15=120$ of these. Once we have these we subtract the invalid combinations The ones which have two hues of the same color. There are three kinds, double orange,green and purple. Which have $\frac{5\cdot4}{2},\frac{7\cdot6}{2}$ and $\frac{4\cdot3}{2},$ so in total we need to substract $37$ cases. And $120-37=83$ as well as in the first solution.
