The number of ways in which any four letters can be selected out of the letters of the word GORCOO is

Options

A. 7
B. 11
C. 15
D. none of these

Answer

To solve the problem of selecting any four letters from the word "GORCOO", we will analyze the letters and their frequencies, and then use combinatorial methods to find the total number of ways to select the letters.

### Step-by-step Solution:

1. **Identify the Letters and Their Frequencies**:
- The letters in "GORCOO" are: G, O, R, C, O, O.
- Frequencies:
- G: 1
- O: 3
- R: 1
- C: 1

2. **Cases of Selection**:
We will consider different cases based on the number of identical letters selected.

**Case 1**: All 4 letters are different.
- The distinct letters are G, O, R, C.
- Since we only have 4 distinct letters, the only way to select 4 different letters is to select all of them.
- Number of ways = 1 (G, O, R, C).

**Case 2**: 2 letters are the same and 2 are different.
- The only letter that can be selected twice is O (since it appears 3 times).
- We need to select 2 more letters from the remaining distinct letters: G, R, C.
- The number of ways to choose 2 letters from G, R, C is given by \( \binom{3}{2} \).
- Calculation:
\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = 3. \]
- Thus, there are 3 ways for this case (O, O, G, R), (O, O, G, C), (O, O, R, C).

**Case 3**: 3 letters are the same and 1 is different.
- Here, we can only select O three times.
- We need to select 1 more letter from the remaining distinct letters: G, R, C.
- The number of ways to choose 1 letter from G, R, C is given by \( \binom{3}{1} \).
- Calculation:
\[ \binom{3}{1} = 3. \]
- Thus, there are 3 ways for this case (O, O, O, G), (O, O, O, R), (O, O, O, C).

3. **Total Number of Ways**:
- Adding all the cases together:
\[ \text{Total} = \text{Case 1} + \text{Case 2} + \text{Case 3} = 1 + 3 + 3 = 7. \]

### Final Answer:
The total number of ways to select any four letters from the word "GORCOO" is **7**.

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