TITLE: Having trouble with a combinatorics question.
QUESTION [5 upvotes]: I'm not so good at combinatorics, but I want to know if my answer for this question is right. Originally this question is written in spanish and it says:

Se dispone de una colección de 30 pelotas divididas en 5 tamaños distintos y 6 colores diferentes de tal manera que en cada tamaño hay los seis colores.¿Cuántas colecciones de 4 pelotas tienen exactamente 2 pares de pelotas del mismo tamaño (que no sean las 4 del mismo tamaño)?.

And here's a translation made by me:

A collection of 30 balls is available, separated in 5 different sizes and 6 different  colors in a way that in each size there are six colors. How many collections  of 4 balls have exactly 2 pairs of balls of same size (which those 4 balls aren't of same size)?

I first wrote this table:
\begin{array}{c|c|c|c|c}
\cdot & Size 1 & Size 2 & Size 3 & Size 4 & Size 5 \\ 
\hline
Color 1 & ① & ❶ & ⒈ & ⑴ & ⓵ \\ 
Color 2 & ② & ❷ & ⒉ & ⑵ & ⓶ \\ 
Color 3 & ③ & ❸ & ⒊ & ⑶ & ⓷ \\ 
Color 4 & ④ & ❹ & ⒋ & ⑷ & ⓸ \\
Color 5 & ⑤ & ❺ & ⒌ & ⑸ & ⓹ \\
Color 6 & ⑥ & ❻ & ⒍ & ⑹ & ⓺
\end{array}
So, an example of a collection of 4 balls that have exactly 2 pairs of balls of the same size is:

①②❶❷

So, for the first column (Size 1) there are 15 combinations of having 2 balls:
①②  ②③  ③④  ④⑤  ⑤⑥
①③  ②④  ③⑤  ④⑥
①④  ②⑤  ③⑥
①⑤  ②⑥
①⑥

Which is the same as: 
$$C_{6}^{2} = \frac{6!}{(6-2)!2!} = 15$$
Or the same as:
$$\sum_{k=1}^{5}k = 15$$
Then, for each row we have 10 combinations:
①❶   ❶⒈  ⒈⑴  ⑴⓵
①⒈  ❶⑴  ⒈⓵
①⑴  ❶⓵  
①⓵

Which is the same as:
$$C_{5}^{2} = \frac{5!}{(5-2)!2!} = 10$$
Or the same as:
$$\sum_{k=1}^{4}k = 10$$
And so, by the rule of product I say that the number of collections of 4 balls having exactly 2 pairs of the same size is:
$$C_{6}^{2} C_{5}^{2} = 150$$
I'll be grateful if someone check my answer and give me further details :D

REPLY [3 votes]: In this kind of problems, the key idea is to transform it into steps and cases in a way that you can use "Addition law" and "Multiplication law". Here you need tow different pairs of balls of the same size, so
Step 1: Select 2 different sizes from those existing 5 sizes;
Step 2: Select 2 balls in each of those 2 selected sizes;
Step 3: Use multiplication law.
And don't forget that "select" means "combination"!