Counting Ternary Strings with Specific Digit Frequencies

Ternary strings, or sequences composed of the digits 0, 1, and 2, are a fascinating subject in combinatorics. This article delves into the method for determining the number of ternary strings with a specific frequency of each digit, providing a detailed explanation with examples and calculations.

Introduction to Ternary Strings

A ternary string is a sequence of digits, each of which can be either 0, 1, or 2. For example, 001220 is a ternary string. When dealing with a ternary string, the goal is to find how many such strings exist that contain a specific number of each digit.

Counting Ternary Strings with 4 Zeros, 2 Ones, and 3 Twos

Consider the question: How many ternary strings are there with exactly 4 zeros (0’s), 2 ones (1’s), and 3 twos (2’s)? This problem can be solved using the concept of permutations of a multiset.

Formula for Permutations of a Multiset

The formula for finding the number of permutations when the multiset contains elements with varying frequencies is given by:

[ text{Number of permutations} frac{n!}{n_1! cdot n_2! cdot n_3!} ]

where:

- ( n ) is the total number of digits in the string. - ( n_1, n_2, n_3 ) are the counts of the distinct digits 0, 1, and 2 respectively.

Step-by-Step Calculation

For the given problem, we have:

- ( n 4 2 3 9 ) (total number of digits in the string). - ( n_0 4 ) (number of zeros). - ( n_1 2 ) (number of ones). - ( n_2 3 ) (number of twos).

Substituting these values into the formula, we get:

[ text{Number of strings} frac{9!}{4! cdot 2! cdot 3!} ]

First, we calculate the factorials:

- ( 9! 362,880 ) - ( 4! 24 ) - ( 2! 2 ) - ( 3! 6 )

Now, substitute these values into the formula:

[ text{Number of strings} frac{362,880}{24 cdot 2 cdot 6} ]

Calculate the denominator:

[ 24 cdot 2 cdot 6 288 ]

Finally, compute the number of strings:

[ text{Number of strings} frac{362,880}{288} 1,260 ]

Thus, the number of ternary strings with exactly 4 zeros, 2 ones, and 3 twos is 1,260.

Alternative Methods and Examples

There are different ways to approach this problem. Here are a couple of examples:

Using the Multinomial Coefficient

The multinomial coefficient is a generalization of the binomial coefficient. The number of ways to arrange the 12 digits (4 zeros, 3 ones, and 5 twos) can be expressed as:

[ frac{12!}{4! cdot 3! cdot 5!} ]

Calculations similar to the previous example can be performed, leading to the same result of 1,260.

Combinatorial Approach

Another method involves using combinations. You can determine the number of ways to place the digits in the string as follows:

- First, choose 4 positions out of 12 for the zeros: ( binom{12}{4} ) - Then, choose 3 positions out of the remaining 8 for the ones: ( binom{8}{3} ) - The twos will occupy the remaining 5 positions: ( binom{5}{5} )

Combining these, the total number of strings can be calculated as:

[ binom{12}{4} cdot binom{8}{3} cdot binom{5}{5} ]

This equals 27,720 ways.

Conclusion

Understanding the number of ternary strings with specific digit frequencies is crucial in combinatorics. By applying the multinomial coefficient and permutations of a multiset, you can solve such problems systematically. The examples provided give a clear insight into how to approach these calculations.

Now, go forth and tackle your ternary string challenges with confidence!