What does it mean to say that a binary code $C$ has length $n$, size $m$ and minimum distance $d$ ? Let $A(n, d)$ be the largest value of $m$ for which there exists an $[n, m, d]$-code. Prove that

$$\frac{2^{n}}{V(n, d-1)} \leqslant A(n, d) \leqslant \frac{2^{n}}{V\left(n,\left\lfloor\frac{1}{2}(d-1)\right\rfloor\right)}$$

where $V(n, r)=\sum_{j=0}^{r}\left(\begin{array}{l}n \\ j\end{array}\right)$.