Question

Companies whose stocks are listed on the NASDAQ stock exchange have their company name represented by either 4 or 5 letters (repetition of letters is allowed). What is the maximum number of companies that can be listed on the NASDAQ?

Short answer

12,338,352

Step-by-step solution

Title - Determine Possible Combinations for 4-Letter Names

Each letter in a 4-letter name can be one of 26 letters (A-Z). So, the number of possible 4-letter combinations is given by: \[ 26^4 \]

Title - Calculate 4-Letter Name Combinations

Calculate the total number of combinations for a 4-letter name: \[ 26^4 = 26 \times 26 \times 26 \times 26 = 456,976 \]

Title - Determine Possible Combinations for 5-Letter Names

Each letter in a 5-letter name can also be one of 26 letters. So, the number of possible 5-letter combinations is given by: \[ 26^5 \]

Title - Calculate 5-Letter Name Combinations

Calculate the total number of combinations for a 5-letter name: \[ 26^5 = 26 \times 26 \times 26 \times 26 \times 26 = 11,881,376 \]

Title - Calculate Total Number of Combinations

Add the number of combinations for both 4-letter and 5-letter names to get the maximum number of companies: \[ 456,976 + 11,881,376 = 12,338,352 \]

Key concepts

letter combinations

In combinatorics, letter combinations refer to different ways letters can be arranged to form words or codes. For this problem, each company name must consist of either 4 or 5 letters, and repetition of letters is allowed. With the English alphabet having 26 letters (A-Z), we can use these letters to form a large variety of names.

For a 4-letter combination, each position (first, second, third, and fourth) can be filled by any of the 26 letters. This can be represented mathematically as a permutation with repetition, written as: \ \[ 26^4 \]

which equals 456,976 combinations. Similarly, for a 5-letter combination, the formula becomes: \ \[ 26^5 \]

resulting in 11,881,376 unique combinations.

stock exchange listings

Stock exchanges such as NASDAQ use ticker symbols (company names) made of letters to identify listed companies. These symbols help in quick identification and trading of stocks. For NASDAQ, ticker symbols can be either 4 or 5 letters long.

Each company's listing is unique, meaning no two companies can have the same symbol. Hence, calculating the total number of possible combinations of these symbols tells us how many companies can be uniquely represented on NASDAQ.

This sort of combinatorial calculation, using permutations and combinations, ensures that the stock exchange can accommodate an extensive range of companies without any symbols overlapping.

permutations

In mathematics, permutations refer to the arrangement of items in a specific order. When repetition is allowed, as is the case with the letter combinations for NASDAQ listings, each position in the sequence can be filled by any of the available choices.

For our current problem, we view each letter in the ticker symbol as a position in a permutation where repetition of letters is allowed.

For example:
\ \[ 26^4 = 26 \times 26 \times 26 \times 26 = 456,976 \onumber \]

The same applies for a 5-letter permutation:
\ \[ 26^5 = 26 \times 26 \times 26 \times 26 \times 26 = 11,881,376 \onumber \]

In permutations with repetition, each element can be used multiple times, offering a vast number of unique combinations.

maximum number of combinations

To find the maximum number of companies that can be listed on the NASDAQ, we need to total the number of possible 4-letter and 5-letter combinations. Calculating each separately and then summing them yields the maximum limit.

Sum of 4-letter combinations: \ \[ 456,976 \]

Sum of 5-letter combinations: \ \[ 11,881,376 \]

Adding these gives: \ \[ 456,976 + 11,881,376 = 12,338,352 \]

This means NASDAQ can have a maximum of 12,338,352 unique company listings. This high number ensures there is ample capacity to uniquely represent a vast number of companies.