Problem 33 - Algebra and Trigonometry

Question

As a financial planner, you are asked to select one stock each from the following groups: 8 Dow Jones stocks 15 NASDAO stocks, and 4 global stocks. How many different portfolios are possible?

Step-by-Step Solution

Answer

480 different portfolios are possible.

Step 1 - Understanding the Problem

We need to determine how many different combinations of one stock each from three groups can be made. These groups are: 8 Dow Jones stocks, 15 NASDAQ stocks, and 4 global stocks.

Step 2 - Using the Fundamental Principle of Counting

The Fundamental Principle of Counting states that if one event can occur in 'm' ways, and a second independent event can occur in 'n' ways, then the two events can occur in 'm x n' ways. This can be extended for more groups.

Step 3 - Calculating the Total Number of Portfolios

Using the principle, the total number of different portfolios is calculated by multiplying the number of possibilities from each group: 8 Dow Jones stocks \times 15 NASDAQ stocks \times 4 global stocks. Mathematically, the equation is: \(8 \times 15 \times 4\).

Step 4 - Performing the Multiplication

Now, perform the multiplication: \(8 \times 15 = 120\) Then multiply the result by the number of global stocks: \(120 \times 4 = 480\).

Step 5 - Conclusion

Thus, the total number of different portfolios possible is 480.

Key Concepts

Combinations

In mathematics, a combination is a way of selecting items from a larger pool where the order does not matter. For instance, if we have a collection of stocks and we want to create a portfolio by selecting one stock from each of several groups, this involves forming combinations. Here, we're interested in how many such portfolios we can form by picking one stock from each of three groups: Dow Jones, NASDAQ, and global stocks. Combinations in this context help us understand the total number of unique portfolios we can create.

Multiplication Principle

The multiplication principle, also known as the Fundamental Principle of Counting, is a basic rule in probability and counting that states if one event can occur in 'm' ways and a second event can occur independently in 'n' ways, then the two events can occur in combination in 'm x n' ways. In the portfolio problem, this principle helps us calculate how many different portfolios can be formed by choosing one stock from each category. In our exercise, there are three groups of stocks: 8 Dow Jones stocks, 15 NASDAQ stocks, and 4 global stocks.

Since the selection of stocks from each group is independent of each other, the total number of possible different portfolios is determined by multiplying the number of options in each group:

• 8 ways to pick a Dow Jones stock
• 15 ways to pick a NASDAQ stock
• 4 ways to pick a global stock

Therefore, the total number of portfolios is given by: \(\text{{Total portfolios}} = 8 \times 15 \times 4 = 480\).

Portfolio Calculation

Portfolio calculation involves determining the number of possible combinations of assets (or stocks) that can be included in a portfolio, based on selection criteria. In this exercise, we needed to compute the total number of different portfolios by selecting one stock from each of the three groups: Dow Jones, NASDAQ, and global stocks.

The calculation proceeds by using the Multiplication Principle:

• Identify the number of choices in each group: 8 Dow Jones stocks, 15 NASDAQ stocks, and 4 global stocks.
• Apply the multiplication principle to determine the total number of combinations: \(8 \times 15 \times 4 = 480\).

Thus, the calculation reveals that there are 480 different possible portfolios we can create by picking one stock from each of the three groups.