A man has 10 friends. In how many ways he can invite one or more of them to a party? a. 10! b. `2^(10)` c. `10!-1` d. `2^(10)-1`

Options

A. 10!
B. `2^(10)`
C. `10!-1`
D. `2^(10)-1`

Answer

To solve the problem of how many ways a man can invite one or more of his 10 friends to a party, we can follow these steps:

### Step 1: Understand the total number of subsets
Each friend can either be invited or not invited. Therefore, for each of the 10 friends, there are 2 choices: invite or not invite.

### Step 2: Calculate the total combinations
Since there are 10 friends, the total number of ways to choose any combination of friends (including the possibility of inviting none) is given by:
\[ 2^{10} \]
This represents all possible subsets of the 10 friends.

### Step 3: Exclude the empty set
However, the problem specifies that the man must invite **one or more** friends. This means we need to exclude the case where no friends are invited (the empty set).

### Step 4: Subtract the empty set
The number of ways to invite one or more friends is:
\[ 2^{10} - 1 \]
This subtraction accounts for the empty set that we do not want to include.

### Step 5: Final answer
Thus, the final answer is:
\[ 2^{10} - 1 \]

### Conclusion
The correct option from the choices provided is:
**d. \(2^{10} - 1\)**

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