12. Arrange $n$ people in a row, and call two adjacent people "friends". If a subset $A$ of the set of these $n$ people satisfies: no two people in $A$ are "friends", then the subset $A$ is called a "bad subset". Try to find the number of "bad subsets" of the set of $n$ people that contain $k$ people.
12. After arranging $n$ people in sequence and numbering them as $1,2,3, \cdots, n$, a "bad subset" of $k$ people actually consists of $k$ non-adjacent natural numbers. If we select a number as an element of the "bad subset" from the sequence $1,2,3,4, \cdots, n$, we insert a "board" after that number. To select $k$ numbers, we need to insert $k$ "boards". The first "board" from the left must have at least 1 number before it, and there must be at least 2 numbers between any two adjacent "boards". There can be no numbers after the $k$-th "board". If we denote the number of numbers in each part from left to right as $x_{1}, x_{2}, x_{3}, \cdots, x_{k}, x_{k+1}$, then $x_{1}+x_{2}+\cdots+x_{k}+x_{k+1}=n$, and $x_{1} \geqslant 1, x_{2} \geqslant 2, x_{3} \geqslant 2, \cdots, x_{k} \geqslant 2, x_{k+1} \geqslant 0$. Clearly, there is a 1-1 correspondence between the set of $k$-element "bad subsets" and the set of integer solutions to the above equation that meet the requirements. Therefore, the number of "bad subsets" is equal to the number of integer solutions to the above equation that meet the requirements, which is $C_{n-k+1}^{k}$.