Number of ways of inviting atleast one people

Question

Find the number of ways of inviting atleast one among 6 people to a party?

Answer 1

A person can invite at least 1 people among 6 people in these manners.

1.  At least 1 people are invited among 6 people. This can be done in ⁶C₁ ways

OR

       2.  2 people can be invited among 6 people. This can be done in ⁶C₂ ways.

OR 

       3.  3 people can be invited among 6 people. This can be done in ⁶C₃ ways

OR

      4.   4 people can be invited among 6 people. This can be done in ⁶C₄ ways.

OR

      5.   5 people can be invited among 6 people. This can be done in ⁶C₅ ways.

OR

      6.   All the 6 people can be invited among 6 people. This can be done in ⁶C₆ ways.

Total number of ways =  ⁶C₁ + ⁶C₂ + ⁶C₃ + ⁶C₄ + ⁶C₅ + ⁶C₆ ways

                                   = 6 + 15 + 20 + 15 + 6+ 1

                                   = 63 ways.

Answer 2

number of ways of inviting atleast one among 6 people to a party

= Total no of ways inviting $ 6$ people $-$Not inviting anyone
$=2^6-1 \\=63$
PS:_ _ _ _ _ _ :$\Rightarrow$ 6 peoples Binary choice $0 \ OR \ 1 \\000000 \rightarrow not  \ inviting\ anyone$