If the coefficient of the middle term in the expansion of ${(1+x)}^{2n+2}$ is p and coefficients of middle term in the expansion of  ${(1+x)}^{2n+1}$ are q and r then

${(1+x)}^{2n+2}$

Coefficient of middle term $={}^{2n+2}{\text{C}}_{n+1}=P$

${(1+x)}^{2n+1}$

Coefficient of middle terms $\frac{{}^{2n+2}{\text{C}}_{2n+1-1}}{2}=q$

$\frac{{}^{2n+2}{\text{C}}_{2n+1+1}}{2}=r$

$\therefore p={}^{2n+2}{\text{C}}_{n+1}$

$q={}^{2n+1}{\text{C}}_n$

$r={}^{2n+1}{\text{C}}_{n+1}$

$q+r={}^{2n+1}{\text{C}}_n+{}^{2n+2}{\text{C}}_{n+1}$

$={}^{2n+2}{\text{C}}_{n+1}$

$\begin{array}{l}=p\\ \therefore q+r=p\end{array}$