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Ta co:$(1+x)^{r}=C^{0}_{r}+C^{1}_{r}x+C^{2}_{r}x^{2}+...+C^{r}_{r}x^{r}$ $(1+x)^{q}=C^{0}_{q}+C^{1}_{q}x+C^{2}_{q}x^{2}+...+C^{q}_{q}x^{q}$He so $x^{p}$ trong tich $(1+x)^{r}(1+x)^{q}la:$ $C^{0}_{r}.C^{p}_{q}+C^{1}_{0}.C^{p-1}_{q}+...+C^{p}_{r}.C^{0}_{q}$ (1)ta lai co:$(1+x)^{r+p}=C^{0}_{r+q}+...+C^{p}_{r+q}x^{p}+...+C^{r+q}_{r+q}.x^{r+q}$He so $x^{p}$ trong $(1+x)^{r+q} $ la $C^{p}_{r+q} (2)$ma $(1+x)^{r}.(1+x)^{q}=(1+x)^{q+r}$nen tu $(1)$ va $(2)$ $\Rightarrow $dpcm

Ta có:$(1+x)^{r}=C^{0}_{r}+C^{1}_{r}x+C^{2}_{r}x^{2}+...+C^{r}_{r}x^{r}$ $(1+x)^{q}=C^{0}_{q}+C^{1}_{q}x+C^{2}_{q}x^{2}+...+C^{q}_{q}x^{q}$Hệ số $x^{p}$ trong tích $(1+x)^{r}(1+x)^{q}là:$ $C^{0}_{r}.C^{p}_{q}+C^{1}_{0}.C^{p-1}_{q}+...+C^{p}_{r}.C^{0}_{q}$ (1)ta lại có:$(1+x)^{r+p}=C^{0}_{r+q}+...+C^{p}_{r+q}x^{p}+...+C^{r+q}_{r+q}.x^{r+q}$Hệ số $x^{p}$ trong $(1+x)^{r+q} $ là $C^{p}_{r+q}$ (2)Mà $(1+x)^{r}.(1+x)^{q}=(1+x)^{q+r} nên từ $(1)$ va $(2)$ \Rightarrow dpcm$

Ta co:$(1+x)^{r}=C^{0}_{r}+C^{1}_{r}x+C^{2}_{r}x^{2}+...+C^{r}_{r}x^{r}$ $(1+x)^{q}=C^{0}_{q}+C^{1}_{q}x+C^{2}_{q}x^{2}+...+C^{q}_{q}x^{q}$He so $x^{p}$ trong tich $(1+x)^{r}(1+x)^{q}la:$ $C^{0}_{r}.C^{p}_{q}+C^{1}_{0}.C^{p-1}_{q}+...+C^{p}_{r}.C^{0}_{q}$ (1)ta lai co:$(1+x)^{r+p}=C^{0}_{r+q}+...+C^{p}_{r+q}x^{p}+...+C^{r+q}_{r+q}.x^{r+q}$He so $x^{p}$ trong $(1+x)^{r+q} $ la $C^{p}_{r+q} (2)ma $(1+x)^{r}.(1+x)^{q}=(1+x)^{q+r} nen tu $(1)$ va $(2)$ $\Rightarrow $dpcm

Ta co:$(1+x)^{r}=C^{0}_{r}+C^{1}_{r}x+C^{2}_{r}x^{2}+...+C^{r}_{r}x^{r}$ $(1+x)^{q}=C^{0}_{q}+C^{1}_{q}x+C^{2}_{q}x^{2}+...+C^{q}_{q}x^{q}$He so $x^{p}$ trong tich $(1+x)^{r}(1+x)^{q}la:$ $C^{0}_{r}.C^{p}_{q}+C^{1}_{0}.C^{p-1}_{q}+...+C^{p}_{r}.C^{0}_{q}$ (1)ta lai co:$(1+x)^{r+p}=C^{0}_{r+q}+...+C^{p}_{r+q}x^{p}+...+C^{r+q}_{r+q}.x^{r+q}$He so $x^{p}$ trong $(1+x)^{r+q} $ la $C^{p}_{r+q} (2)$ma $(1+x)^{r}.(1+x)^{q}=(1+x)^{q+r}$nen tu $(1)$ va $(2)$ $\Rightarrow $dpcm