Math Problem Statement

Find the number of distinct arrangements of the 11 letters in REINTERPRET. Two of the same letter are considered identical (not distinct).

Solution

The problem asks for the number of distinct arrangements of the 11 letters in the word "REINTERPRET", where identical letters are considered indistinguishable.

Step-by-Step Solution:

The word REINTERPRET consists of the following letter frequencies:

  • R: 3 occurrences
  • E: 3 occurrences
  • I: 1 occurrence
  • N: 1 occurrence
  • T: 2 occurrences
  • P: 1 occurrence

The formula for calculating the number of distinct arrangements of letters where some letters repeat is given by:

Total arrangements=11!3!×3!×1!×1!×2!×1!\text{Total arrangements} = \frac{11!}{3! \times 3! \times 1! \times 1! \times 2! \times 1!}

Where:

  • 11!11! is the total number of arrangements of 11 letters.
  • The denominators 3!3!, 3!3!, 1!1!, 1!1!, 2!2!, and 1!1! account for the repeated letters (R, E, T, P, etc.).

Let's calculate it:

11!=3991680011! = 39916800 3!=6,2!=23! = 6, \quad 2! = 2 Denominator=3!×3!×1!×1!×2!×1!=6×6×1×1×2×1=72\text{Denominator} = 3! \times 3! \times 1! \times 1! \times 2! \times 1! = 6 \times 6 \times 1 \times 1 \times 2 \times 1 = 72

Thus, the number of distinct arrangements is:

3991680072=554400\frac{39916800}{72} = 554400

Final Answer:

There are 554,400 distinct arrangements of the letters in "REINTERPRET".

Would you like further details or clarification?

Related Questions:

  1. How do you calculate permutations of words with more than two identical letters?
  2. What if there were no repeating letters in "REINTERPRET"? How would the calculation change?
  3. Can the formula be applied to other words with different numbers of identical letters?
  4. How does the factorial operation scale as the word length increases?
  5. What is the significance of each step in simplifying the arrangement formula?

Tip: Always count the frequencies of each letter carefully before applying the formula to avoid errors in permutation problems.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Combinatorics
Permutations with Repetition

Formulas

n! / (k1! * k2! * ... * km!)

Theorems

Factorial Rule for Permutations with Repetition

Suitable Grade Level

Grades 10-12