How many 7 letter words with or without meaning can be formed out of the letters of the word, “ENCAPSULATION”?

1. Break Down the Problem

To determine how many 7-letter words can be formed from the letters of the word "ENCAPSULATION", we first need to analyze the letters and their occurrences.

2. Relevant Concepts

The word "ENCAPSULATION" consists of the following letters and their frequencies:

• E: 1
• N: 2
• C: 1
• A: 1
• P: 1
• S: 1
• U: 1
• L: 1
• T: 1
• I: 1
• O: 1

Since there are 12 letters in total in "ENCAPSULATION", with the letter N appearing twice, we need to consider cases based on the repetition of letters when forming 7-letter combinations.

3. Analysis and Detail

Case 1: All Letters Different

Choose any 7 letters from the 11 unique letters (E, N, C, A, P, S, U, L, T, I, O).

• The number of ways to choose 7 letters from 11: $\binom{11}{7} = \frac{11!}{7! \cdot (11-7)!} = 330$

• The permutations of these 7 letters: $7! = 5040$

• Total for Case 1: $\text{Total}_1 = 330 \cdot 5040 = 1663200$

Case 2: One Letter Repeated (N)

Choose 6 letters from the remaining 10 unique letters (E, C, A, P, S, U, L, T, I, O).

• The number of ways to choose 6 letters from 10: $\binom{10}{6} = \frac{10!}{6! \cdot (10-6)!} = 210$

• The arrangements of these 6 letters and 1 repeated letter (N): $\frac{7!}{2!} = \frac{5040}{2} = 2520$

• Total for Case 2: $\text{Total}_2 = 210 \cdot 2520 = 529200$

4. Verify and Summarize

Now, we sum the totals from both cases to get the overall number of 7-letter words that can be formed:

$\text{Total} = \text{Total}_1 + \text{Total}_2 = 1663200 + 529200 = 2192400$

Final Answer

Therefore, the total number of 7-letter words that can be formed from the letters of "ENCAPSULATION" is 2,192,400.