A man has 10 friends, in how many ways he can invite one or more of them to a party?

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 Choices
Each of the 10 friends can either be invited or not invited. This gives us 2 choices (invite or not invite) for each friend.

### Step 2: Calculate Total Combinations
Since there are 10 friends, the total number of combinations of inviting or not inviting them is given by:
\[ 2^{10} \]
This includes all possible combinations, including the case where none of the friends are invited.

### Step 3: Exclude the Case of Inviting No One
The problem specifies that we need to invite **one or more** friends. The total combinations calculated in Step 2 includes the scenario where no friends are invited (which is one specific case). Therefore, we need to subtract this case from our total:
\[ \text{Total ways to invite one or more friends} = 2^{10} - 1 \]

### Step 4: Final Calculation
Now we calculate \( 2^{10} \):
\[ 2^{10} = 1024 \]
Thus, the number of ways to invite one or more friends is:
\[ 1024 - 1 = 1023 \]

### Conclusion
The final answer is that the man can invite his friends in **1023 different ways**.

---