A) \[6\]
B) \[5\]
C) \[4\]
D) \[1\]
Correct Answer: B
Solution :
[b] \[\left| \begin{matrix} ^{10}{{C}_{4}} & ^{10}{{C}_{5}} & ^{11}{{C}_{m}} \\ ^{11}{{C}_{6}} & ^{11}{{C}_{7}} & ^{12}{{C}_{m+2}} \\ ^{12}{{C}_{8}} & ^{12}{{C}_{9}} & ^{13}{{C}_{m+4}} \\ \end{matrix} \right|=0\] \[\Rightarrow \left| \begin{matrix} ^{10}{{C}_{4}}{{+}^{10}}{{C}_{5}} & ^{10}{{C}_{5}} & ^{11}{{C}_{m}} \\ ^{11}{{C}_{6}}{{+}^{11}}{{C}_{7}} & ^{11}{{C}_{7}} & ^{12}{{C}_{m+2}} \\ ^{12}{{C}_{8}}{{+}^{12}}{{C}_{9}} & ^{12}{{C}_{9}} & ^{13}{{C}_{m+4}} \\ \end{matrix} \right|=0\] \[({{R}_{1}}\to {{R}_{1}}+{{R}_{2}})\] \[\Rightarrow \left| \begin{matrix} ^{11}{{C}_{5}} & ^{10}{{C}_{5}} & ^{11}{{C}_{m}} \\ ^{12}{{C}_{7}} & ^{11}{{C}_{7}} & ^{12}{{C}_{m+2}} \\ ^{13}{{C}_{9}} & ^{12}{{C}_{9}} & ^{13}{{C}_{m+4}} \\ \end{matrix} \right|=0\] 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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