Question

Dropdi has 5 friends. In how many ways can she invite one or more of them to a dinner?

• A. 31
  
• B. $5^5$
  
• C. 13
  
• D. $2^5$

Answer 1

The correct option is A 31


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 friend can be selected out of 5 in ${}^5{C}_2$ ways
3 friend can be selected out of 5 in ${}^5{C}_3$ ways
4 friend can be selected out of 5 in ${}^5{C}_4$ ways
5 friend 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, ${}^n{C}_1{+}^n{C}_2{+}^n{C}_3+....{+}_n^C=2^n-1$