From a group of 3 freshmen, 4 sophomores, 4 juniors, and 3...

From a group of 3 freshmen, 4 sophomores, 4 juniors, and 3 seniors a committee of size 4 is randomly selected. Find the probability that the committee will consist of
(a) 1 from each class;
(b) 2 sophomores and 2 juniors;
(c) only sophomores or juniors.

Key Concepts

Multiplication Principle

The multiplication principle is used when the selection process involves making a series of independent choices. For example, when forming a committee with one member from each class, the number of ways to create such a committee is the product of the ways to choose one member from each class.

Counting with Restrictions

This concept deals with counting the number of ways to form committees under specific constraints, such as selecting members only from certain classes or requiring a fixed number of members from particular groups. It is crucial for solving problems where not all members of the entire pool are eligible for each spot in the committee.

Combination Formula

This concept involves computing the number of ways to choose a subset of items from a larger set where order does not matter. It is expressed as C(n, k) = n! / (k!(n-k)!), and it is essential for determining both the total number of possible committees and the number of committees that satisfy given conditions.

Probability Calculation

Probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. In the context of committee selection, this involves using combination formulas to count the specific selections that meet the conditions and then dividing by the overall number of ways to form the committee.

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