Find the number of way the books are kept together. The number of ways that $5$ Spanish , $3$ Enligh & $3$ German books are arranged if the books of each language are to be kept together are ... 
My Try : 
$5!+3!+3!$ Which is wrong answer . 
where i am getting wrong ? 
 A: Yes, as others have noted, we need multiplication of the permutations: $$\text{and}\; \iff \;\text{product}, \quad \text{or}\; \iff \;\text{sum}$$
For each one of the $5!$ ways the Spanish books can be arranged, $\Big($ there are $3!$ ways of arranging the English books, and for each of these $5!\cdot 3!$ ways of arranging the Spanish books and English books, there are $\Big($ $3!$ ways of arranging the three German books $\Big)\Big)$ = $5!(3!(3!))$ ways of arranging the books, which will give us the number of arrangements of the books by group: [Spanish books, English books, German books]. 
But we can order these three groups in $3!$ ways, so in total, we have $$3!\Big[5!(3!(3!))\Big] = 3!\cdot 5!\cdot 3!\cdot 3!$$ ways in which the books, overall, can be arranged.
A: suppose one type of books as one item so we can there are $3$ items
so total number of arrangements of $3$ items=$3!$
Now every type of books can be arranged within itself means we put $5$ books of same type together so these $5$ books can be arranged $5!\;$ ways 
so answer=$3!\cdot 5!\cdot 3!\cdot 3!$
A: In enumeration problems like this, you should think of addition as "or" and multiplication as "and". 
With that in mind, $5! + 3! + 3!$ says "arrange the Spanish books or the English books or the German books", which is not appropriate here. We want "and" throughout, which gives $5!3!3!$ ways to arrange the books within each group. 
Finally, we have to arrange the order in which the groups appear, which can be done in $3!$ ways. All together, that gives $5!3!3!3!$ arrangements.
