The value of sum of the series${ }^{14}{C}_0{\cdot }^{15}{C}_1{+}^{14}{C}_1{\cdot }^{15}{C}_2+\dots {+}^{14}{C}_{14}{\cdot }^{15}{C}_{15}$  is 

$\begin{array}{l}(1+x{)}^{14}{=}^{14}{C}_0{+}^{14}{C}_{1}x{+}^{14}{C}_2{x}^2+\dots {+}^{14}{C}_{14}{x}^{14}\\ (x+1{)}^{15}{=}^{15}{C}_0{x}^{15}{+}^{15}{C}_1{x}^{14}{+}^{15}{C}_2{x}^{13}+\dots {+}^{15}{C}_{15}\end{array}$

Multiplying we have 

$(1+x{)}^{29}={\{}^{14}{C}_0{+}^{14}{C}_{1}x{+}^{14}{C}_2{x}^2+\dots {+}^{14}{C}_{14}{x}^{14}\}{\{}^{15}{C}_0{x}^{15}{+}^{15}{C}_1{x}^{14}{+}^{15}{C}_2{x}^{13}+\dots {+}^{15}{C}_{15} \}$

The given expression is coefficient of  $x^{14}$ in the product  $(1+x{)}^{29}$ which will occur in  $T_{r+1}\text{ or }{T}_{14+1}$  

and  is ${ }^{29}{C}_{14}$.