An ice-cream parlour offers only family packs of ice-cream with 11 different flavour s. If each member of a family loves different flavours, then maximum how many such families can purchase the ice-cream if each family contains equal number of persons ?

Options

A. 246
B. 462
C. 123
D. C(11,n)

Answer

To solve the problem, we need to determine how many families can purchase ice cream packs, given that each family member loves different flavors and there are 11 different flavors available.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 11 different flavors of ice cream, and each family member loves a different flavor. We need to find out the maximum number of families that can purchase the ice cream, assuming each family has the same number of members.

2. **Defining Variables**:
Let’s denote the number of members in each family as \( n \). Since each member loves a different flavor, the total number of unique flavors needed for one family of \( n \) members is \( n \).

3. **Finding the Maximum Number of Families**:
Since there are 11 different flavors available, the maximum number of members \( n \) in a family cannot exceed 11 (as each member must love a different flavor). Therefore, the possible values for \( n \) can be any integer from 1 to 11.

4. **Calculating the Number of Families**:
If each family has \( n \) members, the number of families that can be formed with the available flavors is given by the combinations of 11 flavors taken \( n \) at a time, which is represented as \( \binom{11}{n} \).

5. **Finding the Maximum Families**:
To maximize the number of families, we need to consider the maximum value of \( n \). If \( n = 11 \), then there can be only 1 family (since all flavors are used by one family). If \( n = 1 \), then each family can have 1 member, and thus, the maximum number of families would be 11 (since each family can choose a different flavor).

6. **Conclusion**:
Therefore, the maximum number of families that can purchase the ice cream, given that each family contains an equal number of persons, is **11 families** when each family consists of 1 member.

### Final Answer:
**Maximum number of families = 11**