The number of selections of four letters from the letters ofthe word ASSASSINATION is

Options

A. 72
B. 71
C. 66
D. 52

Answer

To solve the problem of selecting four letters from the word "ASSASSINATION," we will analyze the composition of the letters and consider different cases based on the repetition of letters.

### Step-by-Step Solution:

1. **Identify the Letters and Their Frequencies:**
The letters in "ASSASSINATION" and their frequencies are:
- A: 3
- S: 4
- I: 2
- N: 2
- T: 1
- O: 1

2. **Case 1: All Letters are the Same**
- The only letter that can be selected four times is 'A' (4 A's). However, we only have 3 A's, so this case contributes **0** selections.

3. **Case 2: Three Letters are the Same and One is Different**
- The only letter that can be used three times is 'A' (3 A's).
- We can select one additional letter from the remaining letters (S, I, N, T, O).
- There are 5 options (S, I, N, T, O) to choose from.
- Therefore, the number of selections in this case is:
\[ 5C1 = 5 \]
- Total for this case: **5 selections**.

4. **Case 3: Two Letters are the Same and Two are Different**
- We can have two letters the same from A (3 A's), S (4 S's), or I (2 I's), or N (2 N's).
- Let's analyze each sub-case:
- **Sub-case 3.1: 2 A's**
- Choose 2 different letters from S, I, N, T, O (5 letters).
- The number of ways to choose 2 different letters from 5 is:
\[ 5C2 = 10 \]
- **Sub-case 3.2: 2 S's**
- Choose 2 different letters from A, I, N, T, O (5 letters).
- The number of ways to choose 2 different letters from 5 is:
\[ 5C2 = 10 \]
- **Sub-case 3.3: 2 I's**
- Choose 2 different letters from A, S, N, T, O (5 letters).
- The number of ways to choose 2 different letters from 5 is:
\[ 5C2 = 10 \]
- **Sub-case 3.4: 2 N's**
- Choose 2 different letters from A, S, I, T, O (5 letters).
- The number of ways to choose 2 different letters from 5 is:
\[ 5C2 = 10 \]
- Total for this case:
\[ 10 + 10 + 10 + 10 = 40 \]

5. **Case 4: All Letters are Different**
- We need to choose 4 different letters from the available letters (A, S, I, N, T, O).
- There are 6 different letters.
- The number of ways to choose 4 letters from 6 is:
\[ 6C4 = 15 \]

6. **Combine All Cases:**
- Total selections = Case 1 + Case 2 + Case 3 + Case 4
- Total = 0 + 5 + 40 + 15 = **60**.

### Final Answer:
The total number of selections of four letters from the letters of the word "ASSASSINATION" is **66**.