Write down the number of terms in each expression. 2. Are 3x2y3 and -5x3y2 like terms?

The terms are 3, $2ab$, $-4ac$, and $6bc$.
There are $\boxed{\text{4 terms}}$.

b) $6df - 7 + 9b \times 2c$

$9b \times 2c = 18bc$.
The expression becomes $6df - 7 + 18bc$.

The terms are $6df$, $-7$, and $18bc$.
There are $\boxed{\text{3 terms}}$.

c) $(6a - 4b) + (10c + 12d)$

The expression becomes $6a - 4b + 10c + 12d$.

The terms are $6a$, $-4b$, $10c$, and $12d$.
There are $\boxed{\text{4 terms}}$.

d) $(a^2 - d) \times (c^3 - 2d)$

The entire expression is one multiplication operation.
There is $\boxed{\text{1 term}}$.

2. Are $3x^2y^3$ and $-5x^3y^2$ like terms?

a) $3x^2y^3$ and $-5x^3y^2$

For $3x^2y^3$, the variables are $x^2y^3$.
For $-5x^3y^2$, the variables are $x^3y^2$.
Since the exponents for $x$ (2 vs 3) and $y$ (3 vs 2) are not the same, they are not like terms.
The answer is $\boxed{\text{No}}$.

b) $4a^2b^3$ and $-2b^3a^2$

For $4a^2b^3$, the variables are $a^2b^3$.
For $-2b^3a^2$, the variables are $b^3a^2$.
Since $a^2b^3$ is equivalent to $b^3a^2$ (multiplication is commutative), the variables and their exponents are the same.
The answer is $\boxed{\text{Yes}}$.

3. Write down the constant terms in each of these expressions.

a) $6x^7 - 10xy + 5 - 7y^3$

The constant term is 5.
The answer is $\boxed{5}$.

b) $12 + 4abc - 2ba + 13cd$

The constant term is 12.
The answer is $\boxed{12}$.

c) $24m^3n + 12mn^2 - 33 - 60m^2n^3$

The constant term is $-33$.
The answer is $\boxed{-33}$.

4. Simplify where possible.

a) $y^2 + y^2$

The simplified expression is $\boxed{2y^2}$.

b) $3y^2 - 10y^2$

The simplified expression is $\boxed{-7y^2}$.

c) $2x^2 + 9x - x^2 + 12x$

The simplified expression is $\boxed{x^2 + 21x}$.

d) $15ab + ab$

The simplified expression is $\boxed{16ab}$.

e) $3a + 4b - 6c - 2a - 4b + 5c$

The simplified expression is $\boxed{a - c}$.

f) $10x^2y^3 - 5x^2y^3$

The simplified expression is $\boxed{5x^2y^3}$.