Example 1-8: There are 5 Japanese books, 7 English books, and 10 Chinese books. If two books of different languages are taken, how many possibilities are there? If two books of the same language are taken, how many schemes are there? If two books are taken regardless of the language, how many possibilities are there?
(1) Take two books of different languages, according to the multiplication rule:
(1) There are $n_{1}=5 \times 7=35$ possibilities for one English and one Japanese book;
(2) There are $n_{2}=10 \times 5=50$ possibilities for one Chinese and one Japanese book;
(3) There are $n_{3}=10 \times 7=70$ possibilities for one Chinese and one English book.
According to the addition rule, the number of ways to take two books of different languages is
$$
N_{1}=35+50+70=155 \text { ways }
$$
(2) The number of ways to take two books of the same language are:
(1) Both Chinese: $n_{1}^{\prime}=(10 \times 9) / 2=45$,

The first book has 10 possibilities, and the second book has 9 possibilities, but the case where the first book is $a$ and the second book is $b$ is the same as the case where the first book is $b$ and the second book is $a$, so it is counted twice, hence divided by 2.
(2) Both English: $n_{2}^{\prime}=7 \times 6 / 2=21$,
(3) Both Japanese: $n_{3}^{\prime}=5 \times 4 / 2=10$.
According to the addition rule, the number of ways to take two books of the same language is
$$
N_{2}=45+21+10=76 \text { ways }
$$
(3) Without considering whether the languages are the same or not, according to the addition rule
$$
N_{3}=155+76=231 \text { ways, }
$$

or taking two books from 22 books has $22 \times 21 / 2=231$ ways.