Theorem : When the objects are not all distinct, the number of ways to select one or more objects from them is equal to :
$$(q_{1}+1)(q_{2}+1)\cdots (q_{t}+1) -1$$
where, there are $q_{1}$ objects of the 1st kind, $q_{2}$ objects of the second kind, ...$q_{t}$ objects of the t-th kind.
How to prove this theorem? I need hint/resource recommendation.
Let there be $q_1$ objects of first kind, $q_2$ objects of second kind,... and $q_t$ objects of $t^{th}$ kind.
To select at least one object from these, there is always a possibility that you don't select any object from one group, say the $i^{th}$ group ($1≤i≤t$). That's one way of selecting. Again, you may select $1$, $2$, ... or $q_i$ objects from the $i^{th}$ group. That's $q_i$ ways of selecting objects from the $i^{th}$ group. Thus, the total number of ways of selecting at least one object from the $i^{th}$ group is $q_i+1$.
The same procedure applies to groups $1$ through $t$.
Hence, the total number of ways of selecting at least one object from all those groups, is
$N=$ product of ways of selecting from each group $-1$
Therefore,
$$N=(q_1+1)(q_2+1)(q_3+1)...(q_t+1) -1$$
The $-1$ is there to eliminate the way, where you don't choose any object from any group.