Seven nouns, five verbs, and two adjectives are written on a blackboard. We can form a sentence by choosing one word of each type, and we do not care about how much sense the sentence makes. How many ways are there to do this?

Options

A. `7^2 xx 5^2 xx 2^2`
B. `7^1 xx 5^1 xx 2^1 xx 3!`
C. `7! xx 5! xx 2 !`
D. `2^7 xx 2^5 xx 2^2`

Answer

To solve the problem of how many ways we can form a sentence using one noun, one verb, and one adjective from the given words, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number of words available**:
- We have 7 nouns, 5 verbs, and 2 adjectives.

2. **Choose one word from each category**:
- For nouns, we can choose 1 out of 7: **7 ways**.
- For verbs, we can choose 1 out of 5: **5 ways**.
- For adjectives, we can choose 1 out of 2: **2 ways**.

3. **Calculate the total combinations**:
- The total number of ways to choose one word from each category is the product of the number of choices:
\[ \text{Total combinations} = \text{Number of nouns} \times \text{Number of verbs} \times \text{Number of adjectives} \]
\[ = 7 \times 5 \times 2 \]

4. **Perform the multiplication**:
- First, calculate \(7 \times 5 = 35\).
- Then multiply \(35 \times 2 = 70\).
- So, there are **70 ways** to choose one noun, one verb, and one adjective.

5. **Consider the arrangement of the chosen words**:
- Since the order of the words matters in a sentence, we can arrange the three chosen words (noun, verb, adjective) in different ways. The number of arrangements of 3 words is given by \(3!\) (3 factorial).
- Calculate \(3! = 3 \times 2 \times 1 = 6\).

6. **Calculate the final total**:
- Multiply the number of combinations by the number of arrangements:
\[ \text{Final total} = \text{Total combinations} \times \text{Arrangements} \]
\[ = 70 \times 6 = 420 \]

### Final Answer:
Thus, the total number of ways to form a sentence is **420**.

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