数据挖掘导论习题常用数学公式

第6章第5题
Suppose there are $d$ items. We first choose $k$ of the items to form the left-hand side of the rule. There are ${d \choose k}$ ways for doing this. After selecting the items for the left-hand side, there are ${d-k \choose i}$ ways to choose the remaining items to form the right hand side of the rule, where $1 \le i \le d-k$. Therefore the total number of rules $(R)$ is:

$$\begin{align} R &= \sum_{k=1}^d {d \choose k} \sum_{i=1}^{d-k}{d-k \choose i} \\ &= \sum_{k=1}^d {d \choose k}\left( 2^{d-k}-1 \right) \\ &= \sum_{k=1}^d {d \choose k} 2^{d-k} - \sum_{k=1}^d {d \choose k} \\ &= \sum_{k=1}^d {d \choose k} 2^{d-k} - \left[2^d - 1 \right], \end{align}$$
where
$$\sum_{k=1}^d {n \choose i}=2^n -1.$$
Since
$$(1+x)^d= \sum_{i=1}^d {d \choose i}x^{d-i}+x^d,$$
substituting $x=2$ leads to:
$$3^d=\sum_{i=1}^d {d \choose i} 2^{d-i}+2^d.$$
Therefore,the total number of rules is:
$$R=3^d-2^d-\left[ 2^d-1 \right]=3^d-2^{d+1}+1.$$