Ramesh has 6 friends. In how many ways can he invite one or more of them at a dinner

Answer

To solve the problem of how many ways Ramesh can invite one or more of his 6 friends to dinner, we can use the concept of combinations.

### Step-by-Step Solution:

1. **Understanding the Problem**: Ramesh has 6 friends and he can invite any number of them from 1 to 6. We need to calculate the total number of ways he can invite at least one friend.

2. **Using Combinations**: The number of ways to choose \( r \) friends from \( n \) friends is given by the combination formula:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of friends and \( r \) is the number of friends he chooses to invite.

3. **Calculating for Each Case**:
- For \( r = 1 \):
\[ C(6, 1) = \frac{6!}{1!(6-1)!} = 6 \]
- For \( r = 2 \):
\[ C(6, 2) = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]
- For \( r = 3 \):
\[ C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
- For \( r = 4 \):
\[ C(6, 4) = \frac{6!}{4!(6-4)!} = \frac{6 \times 5}{2 \times 1} = 15 \]
- For \( r = 5 \):
\[ C(6, 5) = \frac{6!}{5!(6-5)!} = 6 \]
- For \( r = 6 \):
\[ C(6, 6) = \frac{6!}{6!(6-6)!} = 1 \]

4. **Summing Up the Combinations**: Now, we add up all the combinations for \( r = 1 \) to \( r = 6 \):
\[ \text{Total} = C(6, 1) + C(6, 2) + C(6, 3) + C(6, 4) + C(6, 5) + C(6, 6) \]
\[ \text{Total} = 6 + 15 + 20 + 15 + 6 + 1 = 63 \]

5. **Final Answer**: Therefore, the total number of ways Ramesh can invite one or more of his friends is **63**.