question_answer

The sum  \[\sum\limits_{0\le i\le j\le 10}{{{(}^{10}}{{C}_{  j}})\,{{(}^{j}}{{C}_{i}})}\] is equal to

A) \[{{2}^{10}}-1\]                

B) \[{{2}^{10}}\]                

C) \[{{3}^{10}}-1\] 

D) \[{{3}^{10}}\]

Correct Answer: C

Solution :

[c] \[\sum\limits_{0\le i\le j\le 10}{\left( ^{10}{{C}_{j}} \right)}\,\left( ^{j}{{C}_{i}} \right)\] \[{{=}^{10}}{{C}_{1}}{{(}^{1}}{{C}_{0}}{{+}^{1}}{{C}_{1}}){{+}^{10}}{{C}_{2}}{{(}^{2}}{{C}_{0}}{{+}^{2}}{{C}_{1}}{{+}^{2}}{{C}_{2}})+\] \[....{{+}^{10}}{{C}_{10}}{{(}^{10}}{{C}_{0}}{{+}^{10}}{{C}_{1}}+...{{+}^{10}}{{C}_{10}})\] \[{{=}^{10}}{{C}_{1}}(2){{+}^{10}}{{C}_{2}}({{2}^{2}}){{+}^{10}}{{C}_{3}}({{2}^{3}})+....{{+}^{10}}{{C}_{10}}({{2}^{10}})\] \[={{(2+1)}^{10}}-1={{3}^{10}}-1\]