Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade None
A person wants to invite one or more of his friends for a dinner party. In how many ways can he do so if he has eight friends?
Step-by-Step Solution
Key Concept: Each friend can independently be selected or not selected, creating $2^8$ total combinations, but we must exclude the empty set since at least one friend must be invited.
For each of the 8 friends, the person has 2 choices: invite or don't invite. This gives $2^8$ total ways including inviting no one. Since the person must invite at least one friend, we subtract 1 (the case where nobody is invited), giving $2^8 - 1$ ways. Alternatively, this equals $^8C_1 + ^8C_2 + ^8C_3 + ... + ^8C_8$, which is the sum of all non-empty subsets. Both expressions are equivalent since $2^8 = ^8C_0 + ^8C_1 + ^8C_2 + ... + ^8C_8$.
Correct Answer: 2,4