If '𝑎'  and '𝑏'  are the respective coefficients of $\text{'}{x}^m\text{'}$ and $\text{'}{x}^n\text{'}$ in the expansion of ${\left(1+x\right)}^{m+n}$ then 

$t_{r+1}{=}^n{C}_r\cdot x^{n-r}\cdot y^r$

$\mathrm{Here} n=m+n$

$x=1,y=x$

$\therefore t_{r+1}{=}^{m+n}{C}_r\cdot 1^{m+n-r}\cdot x^r$

${=}^{m+n}{C}_r\cdot x^r$

As per sum

${}^{m+n}C_m=a ,{ }^{m+n}{C}_n=b$

$\Rightarrow {}^{m+n}C_m=a ,{ }^{m+n}{C}_n=b$

$\therefore a=b$