Step 1: Recall the at-least-one rule.
From $n$ distinct objects, the number of non-empty subsets is $2^n-1$, since each object is either picked or not, minus the empty choice.
Step 2: Count Mathematics selections.
With $6$ distinct Mathematics books, the count is $2^6-1=64-1=63$.
Step 3: Count Physics selections.
With $5$ distinct Physics books, the count is $2^5-1=32-1=31$.
Step 4: Count Chemistry selections.
With $4$ distinct Chemistry books, the count is $2^4-1=16-1=15$.
Step 5: Combine with the multiplication principle.
The three choices are independent, so multiply: $63\times 31\times 15$.
Step 6: Multiply out.
$63\times 31=1953$, and $1953\times 15=29295$, option (1).
\[ \boxed{29295} \]