The number of ways in which four letters can be selected from the word degree, is

Options

A. 7
B. 6
C. `(6!)/(3!)`
D. none of these

Answer

To solve the problem of selecting four letters from the word "degree", we can break it down into cases based on the number of times the letter 'E' appears in our selection. The word "degree" consists of the letters D, E, G, R, E, E, which includes three E's and one each of D, G, and R.

### Step-by-Step Solution:

1. **Identify the letters in the word "degree":**
- The letters are D, E, G, R, E, E.
- Unique letters: D, E, G, R.
- Count of letters: D (1), E (3), G (1), R (1).

2. **Case 1: Selecting four letters with exactly one E.**
- We need to choose 3 more letters from D, G, and R.
- The only combination is D, G, R, and E.
- This can be done in **1 way**: {D, G, R, E}.

3. **Case 2: Selecting four letters with exactly two E's.**
- We have two E's fixed, and we need to choose 2 more letters from D, G, and R.
- The combinations can be selected using combinations formula \( \binom{n}{r} \), where n is the total options and r is the number of selections.
- Here, we can choose 2 from D, G, R (which are 3 letters).
- This can be done in \( \binom{3}{2} = 3 \) ways: {E, E, D, G}, {E, E, D, R}, {E, E, G, R}.

4. **Case 3: Selecting four letters with exactly three E's.**
- We have three E's fixed, and we need to choose 1 more letter from D, G, and R.
- This can be done in \( \binom{3}{1} = 3 \) ways: {E, E, E, D}, {E, E, E, G}, {E, E, E, R}.

5. **Total number of ways:**
- Adding all the cases together:
- Case 1: 1 way
- Case 2: 3 ways
- Case 3: 3 ways
- Total = 1 + 3 + 3 = 7.

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

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