A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops of any flavor. The order in which the scoops are purchased is inconsequential. If one wants to purchase 3 scoops of ice-cream, in how many ways can one make that purchase?

Question

A committee of 11 members is to be formed out of 8 males and 5 females. If \( m \) is the number of ways the committee is formed with at least 6 males and \( n \) is the number of ways with at least 3 females, then:

Answer 1

To solve this problem, we can use the "stars and bars" method, which is a popular combinatorial technique used for solving distribution problems. Here, we need to determine the number of ways to distribute 3 scoops of ice-cream among 4 distinct flavors. The order of selecting flavors does not matter, hence this problem is a combination problem.

Consider the following:

Let's denote the 4 flavors as \(A, B, C, \) and \(D\).
We need to find how many ways we can assign the 3 scoops among these 4 flavors.

This is equivalent to finding the number of non-negative integer solutions to the equation:

x_1 + x_2 + x_3 + x_4 = 3

where \(x_1, x_2, x_3, \) and \(x_4\) represent the number of scoops of flavors \(A, B, C, \) and \(D\), respectively.

According to the "stars and bars" theorem, the number of solutions is given by the formula for combinations:

\binom{n + k - 1}{k - 1}

where \(n\) is the total number of scoops, and \(k\) is the number of different flavors.

Here, \(n = 3\) and \(k = 4\), so we need to calculate:

\binom{3 + 4 - 1}{4 - 1} = \binom{6}{3}

Calculating the combination:

\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20

Therefore, there are 20 different ways to distribute the 3 scoops among the 4 flavors.

Thus, the correct answer is 20.

Answer 2