Sum up all of them! (1)

Note that $\dbinom{n}{r} = 0 \ \forall \ r > n$

$\displaystyle \implies \sum_{n=0}^{\infty} \sum_{r=0}^{\infty} \dbinom{n}{r} \dfrac{1}{n!} = \sum_{n=0}^{\infty} \sum_{r=0}^{n} \dbinom{n}{r} \dfrac{1}{n!}$

$\displaystyle = \sum_{n=0}^{\infty} \dfrac{2^n}{n!}$

$= \boxed{e^2}$

$\begin{aligned} A & = \sum_{n=0}^\infty \frac{\begin{pmatrix} n \\ 0 \end{pmatrix}}{n!} + \sum_{n=0}^\infty \frac{\begin{pmatrix} n \\ 1 \end{pmatrix}}{n!} + \sum_{n=0}^\infty \frac{\begin{pmatrix} n \\ 2 \end{pmatrix}}{n!} + \sum_{n=0}^\infty \frac{\begin{pmatrix} n \\ 3 \end{pmatrix}}{n!} + \cdots \\ & = \sum_{n=0}^\infty \frac{1}{n!} + \sum_{n=0}^\infty \frac{n}{n!} + \sum_{n=0}^\infty \frac{n(n-1)}{2!n!} + \sum_{n=0}^\infty \frac{n(n-1)(n-2)}{3!n!} + \cdots \\ & = \sum_{n=0}^\infty \frac{1}{0!n!} + \sum_{n=1}^\infty \frac{1}{1!(n-1)!} + \sum_{n=2}^\infty \frac{1}{2!(n-2)!} + \sum_{n=3}^\infty \frac{1}{3!(n-3)!} + \cdots \\ & = \frac{1}{1!} \sum_{n=0}^\infty \frac{1}{n!} + \frac{1}{2!} \sum_{n=0}^\infty \frac{1}{n!} + \frac{1}{0!} \sum_{n=0}^\infty \frac{1}{n!} + \frac{1}{3!} \sum_{n=0}^\infty \frac{1}{n!} + \cdots \\ & = \sum_{n=0}^\infty \frac{1}{n!} \sum_{n=0}^\infty \frac{1}{n!} \\ & = \left( \sum_{n=0}^\infty \frac{1}{n!}\right)^2 \\ & = e^2 \end{aligned}$

$\implies \lfloor 1000A \rfloor = \lfloor 1000e^2 \rfloor = \boxed{7389}$

Firstly, I would like to mention that if $\displaystyle n<m$ , $\displaystyle \left( \begin{matrix} n \\ m \end{matrix} \right) =0$

Also, $\displaystyle \frac { 1 }{ \left( -1 \right) ! } ,\frac { 1 }{ \left( -2 \right) ! } ,\frac { 1 }{ \left( -3 \right) ! } ,\cdots =0$

$\displaystyle \sum _{ n=0 }^{ \infty }{ \frac { \left( \begin{matrix} n \\ 0 \end{matrix} \right) }{ n! } } +\sum _{ n=0 }^{ \infty }{ \frac { \left( \begin{matrix} n \\ 1 \end{matrix} \right) }{ n! } } +\sum _{ n=0 }^{ \infty }{ \frac { \left( \begin{matrix} n \\ 2 \end{matrix} \right) }{ n! } } +\cdots =\sum _{ n=0 }^{ \infty }{ \frac { \left( \begin{matrix} n \\ 0 \end{matrix} \right) +\left( \begin{matrix} n \\ 1 \end{matrix} \right) +\left( \begin{matrix} n \\ 2 \end{matrix} \right) +\cdots }{ n! } } \\ =\frac { \left( \begin{matrix} 0 \\ 0 \end{matrix} \right) +\left( \begin{matrix} 0 \\ 1 \end{matrix} \right) +\left( \begin{matrix} 0 \\ 2 \end{matrix} \right) +\cdots }{ 0! } +\frac { \left( \begin{matrix} 1 \\ 0 \end{matrix} \right) +\left( \begin{matrix} 1 \\ 1 \end{matrix} \right) +\left( \begin{matrix} 1 \\ 2 \end{matrix} \right) +\cdots }{ 1! } +\cdots \\ =\frac { \left( \begin{matrix} 0 \\ 0 \end{matrix} \right) }{ 0! } +\frac { \left( \begin{matrix} 1 \\ 0 \end{matrix} \right) +\left( \begin{matrix} 1 \\ 1 \end{matrix} \right) }{ 1! } +\frac { \left( \begin{matrix} 2 \\ 0 \end{matrix} \right) +\left( \begin{matrix} 2 \\ 1 \end{matrix} \right) +\left( \begin{matrix} 2 \\ 2 \end{matrix} \right) }{ 2! } +\cdots \quad \quad ,\quad as\quad if\quad n<m,\quad \left( \begin{matrix} n \\ m \end{matrix} \right) =0\\ =\sum _{ n=0 }^{ \infty }{ \frac { \sum _{ k=0 }^{ n }{ \left( \begin{matrix} n \\ k \end{matrix} \right) } }{ n! } }$

In here , we see $\displaystyle \sum _{ n=0 }^{ \infty }{ \frac { \sum _{ k=0 }^{ n }{ \left( \begin{matrix} n \\ k \end{matrix} \right) } }{ n! } } =\sum _{ n=0 }^{ \infty }{ \frac { { 2 }^{ n } }{ n! } } \\ ={ e }^{ 2 }\\$

OR

$\displaystyle \sum _{ n=0 }^{ \infty }{ \frac { \left( \begin{matrix} n \\ 0 \end{matrix} \right) }{ n! } } +\sum _{ n=0 }^{ \infty }{ \frac { \left( \begin{matrix} n \\ 1 \end{matrix} \right) }{ n! } } +\sum _{ n=0 }^{ \infty }{ \frac { \left( \begin{matrix} n \\ 2 \end{matrix} \right) }{ n! } } +\cdots =\sum _{ n=0 }^{ \infty }{ \frac { \frac { n! }{ 0!\times n! } }{ n! } } +\sum _{ n=0 }^{ \infty }{ \frac { \frac { n! }{ 1!\times \left( n-1 \right) ! } }{ n! } } +\sum _{ n=0 }^{ \infty }{ \frac { \frac { n! }{ 2!\times \left( n-2 \right) ! } }{ n! } } +\cdots \\ =\sum _{ n=0 }^{ \infty }{ \frac { 1 }{ 0!\times n! } + } \sum _{ n=0 }^{ \infty }{ \frac { 1 }{ 1!\times \left( n-1 \right) ! } } +\sum _{ n=0 }^{ \infty }{ \frac { 1 }{ 2!\times \left( n-2 \right) ! } } +\cdots \\ =\sum _{ n=0 }^{ \infty }{ \frac { \sum _{ k=1 }^{ \infty }{ \frac { 1 }{ k! } } }{ n! } } ,\quad as\quad \frac { 1 }{ \left( -1 \right) ! } ,\frac { 1 }{ \left( -2 \right) ! } ,\frac { 1 }{ \left( -3 \right) ! } ,\cdots =0$

$\sum _{ n=0 }^{ \infty }{ \frac { e }{ n! } } ={ e }^{ 2 }$