How many tuples? - Part 2

If $T$ has $a$ elements, the number of $b$ -tuples is $\left(2^{b-1}-1\right)^a.$ Here, this is $63^5 = \fbox{992436543}.$

To see this in general, let $P' = P(\{1, \ldots, b-1\}) - \{\varnothing\}$ be the set of all nonempty subsets of $\{1, \ldots, b-1\}.$ Then I claim that there is a bijection between such $b$ -tuples and functions $f : T \to P'.$ This bijection identifies a $b$ -tuple with the function $f(t) = \{i \in \{1, \ldots, b-1\} \colon t \in T_i\}.$

Since $P'$ has $2^{b-1}-1$ elements, the number of functions $f : T \to P'$ is $\left(2^{b-1}-1\right)^a.$

(A more down-to-earth way of stating this proof is: if $T = \{t_1, \ldots, t_5\},$ there are $63$ choices for which sets $t_1$ belongs to, $63$ choices for which sets $t_2$ belongs to, and so on. So there are $63^5$ total possibilities.)