Find the number of distinguishable permutations of the digits of the number 348 838

General
1

find the number of distinguishable permutations of the digits of the number 348 838.

2

How to find the number of distinguishable permutations of the digits of the number 348 838?

Answer:

To find the number of distinguishable permutations of the digits of a number like 348 838, we will follow the principles of combinatorics involving permutations of multiset (elements with repetition). Here’s how we do it step-by-step:

Step 1: Count the Total Digits

The given number is 348 838, which consists of six digits.

Step 2: Identify Frequency of Each Digit

In the number 348 838, the frequency of each digit is as follows:

  • Digit ‘3’ appears 2 times
  • Digit ‘4’ appears 1 time
  • Digit ‘8’ appears 2 times
  • Digit ‘8’ appears 3 times

Step 3: Apply the Permutation Formula for Multisets

The formula for finding the permutations of a multiset is:

\text{Number of permutations} = \frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}

Where:

  • ( n ) is the total number of elements (in this case, digits),
  • ( n_1, n_2, \ldots, n_k ) are the frequencies of the distinguishable elements.

Step 4: Calculate the Permutations

Using the digits from 348 838, apply the formula:

  • Total number of digits, ( n = 6 )
  • Frequency of ‘3’ = 2
  • Frequency of ‘4’ = 1
  • Frequency of ‘8’ = 3

Applying these to the formula, we have:

\text{Number of permutations} = \frac{6!}{2! \times 1! \times 3!}

Now, calculate each factorial:

  • ( 6! = 720 )
  • ( 2! = 2 )
  • ( 1! = 1 )
  • ( 3! = 6 )

Plug these into the equation:

\text{Number of permutations} = \frac{720}{2 \times 1 \times 6} = \frac{720}{12} = 60

Final Answer:

The number of distinguishable permutations of the digits in the number 348 838 is 60.