A) 6
B) 5
C) 4
D) 1
E) 2
Correct Answer: B
Solution :
\[\left| \begin{matrix} ^{10}{{C}_{4}} & ^{10}{{C}_{5}} & ^{11}{{C}_{m}} \\ ^{11}{{C}_{6}} & ^{11}{{C}_{7}} & ^{12}{{C}_{m+2}} \\ ^{12}{{C}_{8}} & ^{12}{{C}_{9}} & ^{13}{{C}_{m+4}} \\ \end{matrix} \right|=0\] Applying\[{{R}_{2}}\to {{R}_{1}}+{{R}_{2}}\] \[\Rightarrow \] \[\left| \begin{matrix} ^{10}{{C}_{4}}{{+}^{10}}{{C}_{5}} & ^{10}{{C}_{5}} & ^{11}{{C}_{m}} \\ ^{11}{{C}_{6}}{{+}^{11}}{{C}_{7}} & ^{11}{{C}_{7}} & ^{12}{{C}_{m+2}} \\ ^{12}{{C}_{8}}{{+}^{12}}{{C}_{9}} & ^{12}{{C}_{9}} & ^{13}{{C}_{m+4}} \\ \end{matrix} \right|=0\] \[\Rightarrow \] \[\left| \begin{matrix} ^{11}{{C}_{5}} & ^{10}{{C}_{5}} & ^{11}{{C}_{m}} \\ ^{12}{{C}_{7}} & ^{11}{{C}_{7}} & ^{12}{{C}_{m+2}} \\ ^{13}{{C}_{9}} & ^{12}{{C}_{9}} & ^{13}{{C}_{m+4}} \\ \end{matrix} \right|=0\] It means either two rows or two columns are identical \[\therefore \] \[^{11}{{C}_{5}}{{=}^{11}}{{C}_{m}}{{,}^{12}}{{C}_{7}}{{=}^{12}}{{C}_{m+2}},\] \[^{13}{{C}_{4}}{{=}^{13}}{{C}_{m+4}}\] \[\Rightarrow \] \[m=5\]
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