Question

A man has 7 friends. In how many ways he can invite one or more of them for a tea party.

• A. 128
• B. 256
• C. 127
• D. 130

Answer 1

The correct option is C

127

He can invite his one friend or two friends or three friends ....... Or all of the seven friends

Selecting one friend out of 7 friends =${}^7{C}_1$

Selecting 2 friends out of 7 friends = ${}^7{C}_2$

.

.

.

.

Selecting all 7 friends = ${}^7{C}_7$

Sum of all these cases = ${}^7{C}_1$ + ${}^7{C}_2$ + .......... + ${}^7{C}_7$ ................................(1)

Add and subtract ${}^7{C}_0$

${}^7{C}_0$ + ${}^7{C}_1$ + ${}^7{C}_2$ + .......... + ${}^7{C}_7$ - ${}^7{C}_0$

Total number of ways = ${}^7{C}_0$ + ${}^7{C}_1$ + ${}^7{C}_2$ + .......... + ${}^7{C}_7$ = $2^7$

case that no friend be invited = ${}^7{C}_0$ which we need to excluded

Required number of ways =$2^7$ - 1 = 127.