Question: 2018 Fall Midterm - 10
Let $n \geq 1$ be an integer and consider a set $\mathcal{B} = \{B_1,B_2,\dots,B_n\}$ of $n$ beer
bottles and a set $\mathcal{C} = \{C_1,C_2,\dots,C_n\}$ of $n$ cider bottles.
For any integer $k$ with $0 \leq k \leq n$, consider subsets $X$ of $\mathcal{B} \cup \mathcal{C}$, such that $X$ consists of exactly $k$ bottles and no two bottles in $X$ have the same index. (For example, if $B_n \in X$, then $C_n \notin X$.)
Let $F(n,k)$ be the number of such subsets X.
Which of the following is true for all integers $n \geq 2$ and $k$ with $1 \leq k \leq n - 1$?
For any integer $k$ with $0 \leq k \leq n$, consider subsets $X$ of $\mathcal{B} \cup \mathcal{C}$, such that $X$ consists of exactly $k$ bottles and no two bottles in $X$ have the same index. (For example, if $B_n \in X$, then $C_n \notin X$.)
Let $F(n,k)$ be the number of such subsets X.
Which of the following is true for all integers $n \geq 2$ and $k$ with $1 \leq k \leq n - 1$?
(a)
$F(n,k) = F(n-1,k)\ +$ $ F(n-1,k-1)$
(b)
$F(n,k) = F(n-1,k)\ +$ $ 2 \cdot F(n-1,k-1)$
(c)
$F(n,k) = F(n,k-1)\ +$ $ F(n-1,k-1)$
(d)
$F(n,k) = F(n,k-1)\ +$ $ 2 \cdot F(n-1,k-1)$