Question

The value of $ \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 $ when m is equal to:

• A. 6
• B. 5
• C. 4
• D. 1
• E. 2

Answer 1

Answer: (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 $ 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 $