How many different arrangements can be obtained from the letters of the word ESSENCE
so that the first two letters are distinct

To determine the number of different arrangements of the letters in the word "ESSENCE" such that the first two letters are distinct, we can follow these steps:

1. **Count the total number of letters**: The word "ESSENCE" has 7 letters.
2. **Identify the letters**: E, S, S, E, N, C, E. Notably, there are 3 E's and 2 S's.
3. **Calculate the total permutations without restrictions**: The total number of permutations of the letters in "ESSENCE" can be calculated considering the repetitions of E and S:
   \[
   \text{Total permutations} = \frac{7!}{3! \cdot 2!} = \frac{5040}{12} = 420
   \]

4. **Consider the restriction that the first two letters must be distinct**:
   - **Case 1**: The first letter is E (and the second letter cannot be E).
   - **Case 2**: The first letter is S (and the second letter cannot be S).
   - **Case 3**: The first letter is N or C (and the second letter can be any of the remaining letters).

We will consider each case separately.

### Case 1: First letter is E
- The remaining letters are: S, S, E, N, C, E
- The second letter cannot be E, so it must be one of the remaining S, S, N, or C.
- Number of choices for the second letter: 4 (S, S, N, C)

For each choice of the first two letters, the remaining 5 letters need to be arranged:
- Total permutations of the remaining 5 letters (S, S, N, C, E): 
  \[
  \text{Permutations} = \frac{5!}{2!} = \frac{120}{2} = 60
  \]

Since there are 4 choices for the second letter, the total for this case is:
  \[
  \text{Case 1 Total} = 4 \times 60 = 240
  \]

### Case 2: First letter is S
- The remaining letters are: E, S, E, N, C, E
- The second letter cannot be S, so it must be one of the remaining E, E, N, or C.
- Number of choices for the second letter: 4 (E, E, N, C)

For each choice of the first two letters, the remaining 5 letters need to be arranged:
- Total permutations of the remaining 5 letters (E, S, E, N, C): 
  \[
  \text{Permutations} = \frac{5!}{2!} = \frac{120}{2} = 60
  \]

Since there are 4 choices for the second letter, the total for this case is:
  \[
  \text{Case 2 Total} = 4 \times 60 = 240
  \]

### Case 3: First letter is N or C
- There are 2 choices for the first letter (N or C)
- The remaining letters need to be arranged: E, S, S, E, N/C, E
- The second letter can be any of the remaining letters (except the first letter itself), so there are 6 options for the second letter.

For each arrangement of the first two letters, the remaining 5 letters need to be arranged:
- Total permutations of the remaining 5 letters: 
  \[
  \text{Permutations} = \frac{5!}{3! \cdot 2!} = \frac{120}{12} = 10
  \]

Since there are 6 choices for the second letter, and 2 choices for the first letter, the total for this case is:
  \[
  \text{Case 3 Total} = 2 \times 6 \times 10 = 120
  \]

### Total Number of Valid Arrangements
\[
\text{Total Valid Arrangements} = \text{Case 1 Total} + \text{Case 2 Total} + \text{Case 3 Total} = 240 + 240 + 120 = 600
\]

So, the number of different arrangements of the letters in "ESSENCE" where the first two letters are distinct is \( \boxed{600} \).

Would you like more details or have any questions? Here are some questions you might want to ask:

1. How do you handle cases with multiple restrictions?
2. What if the word had more or fewer repeated letters?
3. Can you explain more about permutations with repetition?
4. How do you determine the factorial of a number?
5. Can you give an example with a different word?

**Tip:** When dealing with permutations with restrictions, consider breaking the problem into distinct cases and calculating the number of valid arrangements for each case before combining the results.

Explore how to calculate the number of different arrangements from the letters of the word ESSENCE where the first two letters are distinct. Understand the permutations involved and the combinatorial calculations required. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation