Problem 39 - Algebra and Trigonometry

Question

Companies whose stocks are listed on the New York Stock Exchange (NYSE) have their company name represented by \(1,2,\) or 3 letters (repetition of letters is allowed). What is the maximum number of companies that can be listed on the NYSE?

Step-by-Step Solution

Answer

The maximum number of companies that can be listed is 18,278.

Step 1: Consider the Number of Letters in Each Company Name

Companies can have their names represented by 1, 2, or 3 letters.

Step 2: Calculate the Possible Combinations for 1-Letter Names

Since there are 26 letters in the English alphabet, the total number of 1-letter combinations is 26.

Step 3: Calculate the Possible Combinations for 2-Letter Names

Each of the two letters can independently be one of 26 letters. Thus, the number of 2-letter combinations is calculated as follows: \[ 26 \times 26 = 676 \]

Step 4: Calculate the Possible Combinations for 3-Letter Names

Similarly, each of the three letters can independently be one of 26 letters. Thus, the number of 3-letter combinations is calculated by: \[ 26 \times 26 \times 26 = 26^3 = 17576 \]

Step 5: Sum All Possible Combinations

Now sum the possible combinations for 1-letter, 2-letter, and 3-letter names to find the total maximum number of companies: \[ 26 + 676 + 17576 = 18278 \]

Key Concepts

permutations

In permutations, we're concerned with the different ways we can arrange elements given a specific set. For example, if we have three letters, A, B, and C, different ways like ABC, ACB, BAC, etc., are unique permutations. This concept is vital when understanding NYSE stock symbols because arrangements of letters matter. For combinatorial problems, permutations help us calculate exact numbers.

alphabetic combinations

Alphabetic combinations involve using letters to form different sequences. In the case of NYSE stock symbols, names can have 1, 2, or 3 letters. To understand combinations, think of creating every possible arrangement of letters for each combination length. For example:

• 1-letter combinations: A, B, C, ..., Z.
• 2-letter combinations: AA, AB, AC, ..., ZZ.
• 3-letter combinations: AAA, AAB, AAC, ..., ZZZ.

By calculating these, we sum up to find the total possible combinations.

NYSE stock symbols

NYSE stock symbols represent companies with either 1, 2, or 3 letters, enabling easy recognition like A for Agilent or IBM for International Business Machines. We calculate the maximum number of combinations by:

• 1-letter symbols: 26 options (A-Z)
• 2-letter symbols: 26 x 26= 676 options
• 3-letter symbols: 26 x 26 x 26= 17576 options

Summing all, we get the total capacity for NYSE stock symbols:

26 + 676 + 17576 = 18278.
Understanding these combinations helps us appreciate how NYSE efficiently uses limited letters to create diverse stock symbols.