(1)

\begin{align*} \sum_{k=0}^{n}C(n,k) & =\sum_{k=0}^{n}C(n,k)1^{k}1^{n-k}\\ & =(1+1)^{n}\\ & =2^{n} \end{align*}

(2)

\begin{align*} \sum_{k=0}^{n}P(k,m)C(n,k) & =\sum_{k=0}^{n}\frac{n!}{(k-m)!(n-k)!}\\ & =\frac{n!}{(n-m)!}\sum_{k=0}^{n}\frac{(n-m)!}{(k-m)!(n-k)!}\\ & =P(n,m)\sum_{k=0}^{n}C(n-m,n-k)\\ & =P(n,m)\sum_{k=0}^{n-m}C(n-m,k)\\ & =P(n,m)2^{n-m} \end{align*}

(3)

\begin{align*} \sum_{k=0}^{n}k^{m}C(n,k) & =n\sum_{k=0}^{n}k^{m-1}C(n-1,k-1)\\ & =n\sum_{k=0}^{n-1}(k+1)^{m-1}C(n-1,k) \end{align*}

(4)

\begin{align*} \sum_{k=0}^{n}k^{m}C(n,k) & =\sum_{k=0}^{n}\sum_{j=0}^{m}S_{2}(m,j)P(k,j)C(n,k)\\ & =\sum_{j=0}^{m}S_{2}(m,j)P(n,j)2^{n-j} \end{align*}

(5)

\begin{align*} \sum_{k=0}^{n}kC(n,k) & =\sum_{j=0}^{1}S_{2}(1,j)P(n,j)2^{n-j}\\ & =S_{2}(1,0)P(n,0)2^{n-0}+S_{2}(1,1)P(n,1)2^{n-1}\\ & =n2^{n-1} \end{align*}

(5)-2

\begin{align*} \sum_{k=0}^{n}kC(n,k) & =\sum_{k=0}^{n}nC(n-1,k-1)\\ & =n\sum_{k=1}^{n}C(n-1,k-1)\\ & =n\sum_{k=0}^{n-1}C(n-1,k)\\ & =n2^{n-1} \end{align*}

(5)-3

\begin{align*} \sum_{k=0}^{n}kC(n,k) & =\left[\frac{d}{dx}\sum_{k=0}^{n}C(n,k)x^{k}\right]_{x=1}\\ & =\left[\frac{d}{dx}(1+x)^{n}\right]_{x=1}\\ & =\left[n(1+x)^{n-1}\right]_{x=1}\\ & =n2^{n-1} \end{align*}

(6)

\begin{align*} \sum_{k=0}^{n}k^{2}C(n,k) & =\sum_{j=0}^{2}S_{2}(2,j)P(n,j)2^{n-j}\\ & =S_{2}(2,0)P(n,0)2^{n-0}+S_{2}(2,1)P(n,1)2^{n-1}+S_{2}(2,2)P(n,2)2^{n-2}\\ & =n2^{n-1}+n(n-1)2^{n-2}\\ & =(n^{2}+n)2^{n-2} \end{align*}

(6)-2

\begin{align*} \sum_{k=0}^{n}k^{2}C(n,k) & =\sum_{k=0}^{n}\left\{ k(k-1)C(n,k)+kC(n,k)\right\} \\ & =\sum_{k=0}^{n}\left\{ n(n-1)C(n-2,k-2)+nC(n-1,k-1)\right\} \\ & =n(n-1)\sum_{k=2}^{n}C(n-2,k-2)+n\sum_{k=1}^{n}C(n-1,k-1)\\ & =n(n-1)\sum_{k=0}^{n-2}C(n-2,k-2)+n\sum_{k=0}^{n-1}C(n-1,k-1)\\ & =n(n-1)2^{n-2}+n2^{n-1}\\ & =(n^{2}+n)2^{n-2} \end{align*}

(6)-3

\begin{align*} \sum_{k=0}^{n}k^{2}C(n,k) & =\left[\frac{d}{dx}x\frac{d}{dx}\sum_{k=0}^{n}C(n,k)x^{k}\right]_{x=1}\\ & =\left[\frac{d}{dx}x\frac{d}{dx}(1+x)^{n}\right]_{x=1}\\ & =n\left[\frac{d}{dx}x(1+x)^{n-1}\right]_{x=1}\\ & =n\left[\frac{d}{dx}\left\{ (1+x)^{n}-(1+x)^{n-1}\right\} \right]_{x=1}\\ & =n\left[n(1+x)^{n-1}-(n-1)(1+x)^{n-2}\right]_{x=1}\\ & =n\left\{ n2^{n-1}-(n-1)2^{n-2}\right\} \\ & =(n^{2}+n)2^{n-2} \end{align*}