Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the number of distinguishable permutations of the group of letters. \(\mathbf{B}, \mathbf{B}, \mathbf{B}, \mathbf{T}, \mathbf{T}, \mathbf{T}, \mathbf{T}, \mathbf{T}\)

Step-by-Step Solution

Answer

The number of distinguishable permutations of these letters is \(\frac{8!}{3! * 5!}\).

Step 1: Identify the elements and their occurrences

Here, we have the letter 'B' appearing 3 times and the letter 'T' appearing 5 times. So we write this down as n = 3 for B and m = 5 for T. The total number of elements is then 3 + 5 = 8.

Step 2: Calculate the total number of permutations

The total number of permutations without considering the repetitions is given by the factorial of the total number of elements. Factorial, denoted as '!', means multiplying all positive integers up to that number. So, the total number of permutations is \(8!\).

Step 3: Calculate the number of repeated permutations

We then calculate the number of permutations for the repeated elements. This is done by taking the factorial of the number of times each element occurs. So, we have \(3!\) for 'B' and \(5!\) for 'T'.

Step 4: Calculate the number of distinguishable permutations

To get the number of distinguishable permutations, we use the formula for permutations of a multiset, which says to divide the total number of permutations by the product of the number of repeated permutations. So we divide \(8!\) by \((3! * 5!)\))