A man has 6 friends . In how many ways ways may be invite one or more of them to dinner?

Answer

To solve the problem of how many ways a man can invite one or more of his 6 friends to dinner, we can use the concept of combinations. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
The man can invite any number of friends from 1 to 6. We need to calculate the total number of ways he can do this.

### Step 2: Calculate Combinations for Each Case
We will calculate the number of ways to invite 1 friend, 2 friends, 3 friends, 4 friends, 5 friends, and all 6 friends.

1. **Inviting 1 friend**: The number of ways to choose 1 friend from 6 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of friends and \( r \) is the number of friends to choose.
\[ \binom{6}{1} = 6 \]

2. **Inviting 2 friends**:
\[ \binom{6}{2} = 15 \]

3. **Inviting 3 friends**:
\[ \binom{6}{3} = 20 \]

4. **Inviting 4 friends**:
\[ \binom{6}{4} = 15 \]

5. **Inviting 5 friends**:
\[ \binom{6}{5} = 6 \]

6. **Inviting all 6 friends**:
\[ \binom{6}{6} = 1 \]

### Step 3: Sum the Combinations
Now, we will add all these combinations together to find the total number of ways to invite one or more friends.
\[ \text{Total} = \binom{6}{1} + \binom{6}{2} + \binom{6}{3} + \binom{6}{4} + \binom{6}{5} + \binom{6}{6} \]
\[ \text{Total} = 6 + 15 + 20 + 15 + 6 + 1 = 63 \]

### Step 4: Conclusion
The total number of ways the man can invite one or more of his 6 friends to dinner is **63**.

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