I assume all sets are finite, though the argument below can be extended to deal with the infinite case.
$\bullet$ The sume rule essentially states that $|A \cup B| = |A| + |B| - |A \cap B|$. How do you show that if $A \cap B = \emptyset$, then $|A \cup B| = |A| + |B|$?
This is accomplished by simply noticing that if $A = \{x_1, \dots , x_n\}$ and $B = \{y_1, \dots , y_m\}$, then $A \cup B = \{x_1, \dots , x_n, y_1, \dots , y_m\}$. Note also that $x_i \neq y_j$ for any pair $(i,j)$. Hence, $A \cup B$ is a set with $n + m$ distinct elements, and so $|A \cup B| = n + m = |A| + |B|$.
What if $A \cap B \neq \emptyset$? Then, write $A \cup B = (A^c \cap B) \cup (A \cap B) \cap (B^c \cap A)$ which is a union of disjoint sets and apply the above argument to these disjoint sets.
$\bullet$ The product rule essentially states that $|A \times B| = |A| |B|$. This follows essentially as before. Write out $A \times B$, and find the cardinality of the set.
Once you establish the sum-rule and product-rule for two sets, use induction for the general case.
Best Answer
Let $P(m)$ be the following statement:
$P(1)$ is obvious, you’re given that $P(2)$ is true, and you’re to prove that $P(m)$ is true for all positive integers $m$. This is a natural setting for a proof by induction, and you’ve been handed the basis step for free. Thus, all you have to do is prove the induction step: if $P(m)$ is true, then so is $P(m+1)$. In words, if the product rule holds for $m$ tasks, then it holds for $m+1$ tasks. So assume that $P(m)$ is true for some $m\ge 2$; we’ll try to prove $P(m+1)$.
To that end suppose that we have $m+1$ tasks, and that task $T_k$ can be performed in $n_k$ ways, where $k=1,\dots,m+1$. We’d like to show that the entire set of $m+1$ tasks can be performed in $n_1n_2\dots n_mn_{m+1}$ ways.
HINT: Think of the combination of tasks $T_1$ through $T_m$ as a single very large task; call that task $T$. To complete tasks $T_1$ through $T_{m+1}$ you must complete the two tasks $T$ and $T_{m+1}$. You’re assuming that the product rule is true for $m$ tasks, so how many ways are there to perform task $T$? How many ways are there to perform the pair of tasks $T$ and $T_{m+1}$?