$7$ nouns, $5$ verbs, and $2$ adjectives are written on a blackboard. We can form a sentence by choosing $1$ from each available set in any order. Without caring it makes sense or not, what is the number of ways of doing this?
If we use a permutation and just do $7 \cdot 5 \cdot 2$, it gives us $70$. But the problem states that the sentence can be in any order. I am confused exactly how I am supposed to figure out the number of possibilities. Would you multiply $70$ by the number of possible arrangements? If so, what is that number?
Given a noun, verb, and adjective, there are three ways to select the first word of the sentence, which leaves two ways to select the second word of the sentence, and one way to select the third word of the sentence. Thus, the three selected words can be arranged in $3! = 3 \cdot 2 \cdot 1 = 6$ ways. Since there are seven ways of selecting the noun, five ways of selecting the verb, and two ways of selecting the adjective, the number of admissible sentences is $7 \cdot 5 \cdot 2 \cdot 3! = 420$.