In how many ways can I invite one or more six friends to a dinner?

Answer

To solve the problem of how many ways you can invite one or more of your six friends to dinner, we can use the concept of combinations. Here’s a step-by-step solution:

### Step 1: Understand the Problem
You have 6 friends and you want to invite at least one of them. This means you can invite any number from 1 to 6 friends.

### Step 2: Use Combinations
The number of ways to choose k friends from a total of n friends is given by the combination formula:
\[ nCk = \frac{n!}{k!(n-k)!} \]
where \( n \) is the total number of friends and \( k \) is the number of friends you want to invite.

### Step 3: Calculate Combinations for Each Case
You need to calculate the combinations for inviting 1 friend, 2 friends, 3 friends, 4 friends, 5 friends, and all 6 friends:
- For 1 friend: \( 6C1 \)
- For 2 friends: \( 6C2 \)
- For 3 friends: \( 6C3 \)
- For 4 friends: \( 6C4 \)
- For 5 friends: \( 6C5 \)
- For 6 friends: \( 6C6 \)

### Step 4: Calculate Each Combination
Now we will calculate each of these:
- \( 6C1 = 6 \)
- \( 6C2 = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \)
- \( 6C3 = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \)
- \( 6C4 = \frac{6!}{4!(6-4)!} = \frac{6 \times 5}{2 \times 1} = 15 \)
- \( 6C5 = 6 \)
- \( 6C6 = 1 \)

### Step 5: Sum All Combinations
Now, add all these combinations together to find the total number of ways to invite one or more friends:
\[ Total = 6C1 + 6C2 + 6C3 + 6C4 + 6C5 + 6C6 = 6 + 15 + 20 + 15 + 6 + 1 = 63 \]

### Final Answer
Thus, the total number of ways to invite one or more of your six friends to dinner is **63**.

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