# Choosing two colours of different tones.

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.

• does orange and orange count? Commented Jun 1, 2014 at 23:45
• @user3678068 I don't think so. The OP says different $tones$
– Joao
Commented Jun 2, 2014 at 0:07
• @user3678068: No, it doesn't count. Commented Jun 2, 2014 at 0:08

## 1 Answer

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.