c
In series combination \(\Delta l =l_{1}+l_{2}\)

\(Y =\frac{ F / A }{\Delta l / l} \Rightarrow \Delta l=\frac{ F l}{ AY }\)

\(\Rightarrow \Delta l \propto \frac{l}{ Y }\)

Equivalent length of rod after joining is \(=2 l\)

As, lengths are same and force is also same in series

\(\Delta l=\Delta l_{1}+\Delta l_{2}\)

\(\frac{l \text { eq }}{ Y _{\text {eq }}}=\frac{l}{ Y _{1}}+\frac{l}{ Y _{2}}\) \(\Rightarrow \frac{2 l}{ Y }=\frac{l}{ Y _{1}}+\frac{l}{ Y _{2}}\)

\(\therefore Y=\frac{2 Y_{1} Y_{2}}{Y_{1}+Y_{2}}\)