A) \[{{3}^{5}}-{{2}^{5}}+1\]
B) \[{{3}^{5}}-{{2}^{5}}\]
C) \[{{3}^{5}}-1\]
D) \[{{2}^{5}}-1\]
Correct Answer: B
Solution :
[b] \[\sum\limits_{0\le i<j\le 5}{\sum{\left( ^{5}{{C}_{j}} \right)\,\,\left( ^{j}{{C}_{i}} \right)}}\] \[{{=}^{5}}{{C}_{1}}{{.}^{1}}{{C}_{0}}{{+}^{5}}{{C}_{2}}{{(}^{2}}{{C}_{0}}{{+}^{2}}{{C}_{1}})+...{{+}^{5}}{{C}_{5}}{{(}^{5}}{{C}_{0}}{{+}^{5}}{{C}_{1}}+...{{+}^{5}}{{C}_{4}})\] \[{{=}^{5}}{{C}_{1}}({{2}^{1}}-1){{+}^{5}}{{C}_{2}}({{2}^{2}}-1)+....{{+}^{5}}{{C}_{5}}({{2}^{5}}-1)\] \[={{(}^{5}}{{C}_{1}}{{.2}^{1}}{{+}^{5}}{{C}_{2}}{{.2}^{2}}+...{{+}^{5}}{{C}_{5}}{{.2}^{5}})-{{(}^{5}}{{C}_{1}}{{+}^{5}}{{C}_{2}}+...{{+}^{5}}{{C}_{5}})\]\[[{{(1+2)}^{5}}-1]-[{{2}^{5}}-1]\] \[={{3}^{5}}-{{2}^{5}}\]
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