A) \[31\]
B) \[{{5}^{5}}\]
C) \[13\]
D) \[{{2}^{5}}\]
E) None of these
Correct Answer: A
Solution :
Explanation Option [a] is correct. She may invite one or more friends by selecting either 1 or 2 or 3 or 4 or 5 friends out of 5 friends. \[\therefore \] 1 friend can be selected out of 5 in \[^{5}{{C}_{1}}\] ways 2 friends can be selected out of 5 in \[^{5}{{C}_{2}}\] ways 3 friends can be selected out of 5 in \[^{5}{{C}_{3}}\] ways 4 friends can be selected out of 5 in \[^{5}{{C}_{4}}\] ways 5 friends can be selected out of 5 in \[^{5}{{C}_{5}}\] ways Hence the required number of ways \[{{=}^{5}}{{C}_{1}}{{+}^{5}}{{C}_{2}}{{+}^{5}}{{C}_{3}}{{+}^{5}}{{C}_{4}}{{+}^{5}}{{C}_{5}}\] \[=5+10+10+5+1=31\] Alternatively: \[^{5}{{C}_{1}}+{{\,}^{5}}{{C}_{2}}+{{\,}^{5}}{{C}_{3}}+{{\,}^{5}}{{C}_{4}}+{{\,}^{5}}{{C}_{5}}={{2}^{5}}-1=31\] since,\[^{n}{{C}_{1}}+{{\,}^{n}}{{C}_{2}}+{{\,}^{n}}{{C}_{3}}+\,.....+{{\,}^{n}}{{C}_{n}}={{2}^{n}}-1\]
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