Merit Badge Requirements To complete the Citizenship in the World merit badge, one must select two of the following seven organizations and describe their role in the world. Source: Boy Scouts of America
1. The United Nations
2. The World Court
3. World Organization of the Scout Movement
4. The World Health Organization
5. Amnesty International
6. The International Committee of the Red Cross
7. CARE
Comment on the likelihood that the pair “The United Nations” and “Amnesty International” will be selected.

Verified step by step guidance

1. Identify the total number of organizations available to choose from, which is 7 in this case.
2. Understand that the problem involves selecting 2 organizations out of the 7, so the total number of possible pairs is given by the combination formula \(\displaystyle C(n, k) = \frac{n!}{k!(n-k)!}\), where \(n=7\) and \(k=2\).
3. Calculate the total number of possible pairs using the formula: \(\displaystyle C(7, 2) = \frac{7!}{2! \times 5!}\).
4. Recognize that the pair “The United Nations” and “Amnesty International” is just one specific pair out of all possible pairs.
5. Determine the probability of selecting this specific pair by dividing 1 (the favorable outcome) by the total number of possible pairs calculated in step 3, i.e., \(\displaystyle P = \frac{1}{C(7, 2)}\).

Key Concepts

Probability and Likelihood

Probability measures how likely an event is to occur, expressed as a number between 0 and 1. Understanding likelihood helps assess the chance of selecting a specific pair from a set, considering all possible combinations equally or with given preferences.

Combinatorics and Counting Principles

Combinatorics involves counting the number of ways to select items from a set. For example, choosing 2 organizations out of 7 involves combinations, calculated using formulas to determine total possible pairs and the probability of any specific pair.

Contextual Factors Affecting Selection

Beyond pure mathematics, real-world factors like the prominence or relevance of organizations influence selection likelihood. Understanding these qualitative aspects helps interpret probabilities in practical scenarios, such as merit badge requirements.