Problem 21

Given $card(A) = M$ and $card(B) = N$ with $M < N$, and considering $A \subseteq C \subseteq B$, we aim to find the number of possible sets $C$.

1. **Identify the relationship between $A$ and $B$**: Since $A \subseteq B$, every element of $A$ is also an element of $B$. This implies $card(\complement_{B}A) = N - M$, where $\complement_{B}A$ represents the elements in $B$ that are not in $A$.

2. **Understand the constraints on $C$**: The condition $A \subseteq C \subseteq B$ means that $C$ must include all elements of $A$ and may include any (including none or all) of the elements from $\complement_{B}A$.

3. **Calculate the number of possible sets $C$**: The number of subsets of $\complement_{B}A$ (which represents the elements that can be added to $A$ to form $C$) is $2^{card(\complement_{B}A)} = 2^{N-M}$. This is because each element of $\complement_{B}A$ can either be included in or excluded from a subset, leading to two choices per element.

Therefore, the number of sets $C$ that meet the conditions is $2^{N-M}$.

Final answer: $\boxed{2^{N-M}}$.