Write the number of terms and also list out the terms in each of the following algebraic expressions.
`(ab) (bc)(ca)- (ab) a+(bc) b -(a^2)b -(b^2)b`

Answer

To solve the problem, we need to simplify the given algebraic expression and identify the number of terms along with listing them out. The expression we have is:

\[ (ab)(bc)(ca) - (ab)a + (bc)b - (a^2)b - (b^2)b \]

### Step 1: Simplify each term
1. **First Term:**
\[ (ab)(bc)(ca) = a \cdot b \cdot b \cdot c \cdot c \cdot a = a^2b^2c^2 \]

2. **Second Term:**
\[ -(ab)a = -a^2b \]

3. **Third Term:**
\[ (bc)b = b^2c \]

4. **Fourth Term:**
\[ -(a^2)b = -a^2b \]

5. **Fifth Term:**
\[ -(b^2)b = -b^3 \]

### Step 2: Combine the simplified terms
Now, we can combine all these simplified terms:
\[ a^2b^2c^2 - a^2b + b^2c - a^2b - b^3 \]

### Step 3: Combine like terms
Notice that \(-a^2b\) appears twice:
\[ a^2b^2c^2 - 2a^2b + b^2c - b^3 \]

### Step 4: Identify the terms
Now, we can identify the distinct terms in the expression:
1. \(a^2b^2c^2\)
2. \(-2a^2b\)
3. \(b^2c\)
4. \(-b^3\)

### Conclusion
The final expression is:
\[ a^2b^2c^2 - 2a^2b + b^2c - b^3 \]

### Number of Terms
The number of terms in the expression is **4**.

### List of Terms
The terms are:
1. \(a^2b^2c^2\)
2. \(-2a^2b\)
3. \(b^2c\)
4. \(-b^3\)

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