From the string abacabababcdced, if 5 letters should be selected , then the number of ways in which this selection can be done is

Answer

To solve the problem of selecting 5 letters from the string "abacabababcdced", we will break down the solution into cases based on the frequency of the letters.

### Step-by-Step Solution:

1. **Identify the Frequency of Each Letter**:
- From the string "abacabababcdced", we count the occurrences of each letter:
- A: 5 times
- B: 4 times
- C: 3 times
- D: 2 times
- E: 1 time

2. **Define Cases for Selection**:
We will consider different cases based on how many letters are the same.

**Case 1**: All 5 letters are the same.
- The only option is "aaaaa".
- Number of ways = 1.

**Case 2**: 4 letters are the same, and 1 letter is different.
- Possible combinations: "aaaaX" where X can be B, C, D, or E.
- Choices for X: B (1 way), C (1 way), D (1 way), E (1 way).
- Total = 4 ways.

**Case 3**: 3 letters are the same, and 2 letters are different.
- Possible combinations: "aaaXY".
- Choices for A: A, B, C (since they appear at least 3 times).
- If we select A, then X and Y can be chosen from B, C, D, E.
- Total combinations = 3 (for A, B, C) * (number of ways to choose 2 different letters from remaining letters).
- Remaining letters after choosing A: B, C, D, E (4 letters).
- Number of ways to choose 2 different letters from 4 = C(4, 2) = 6.
- Total = 3 * 6 = 18 ways.

**Case 4**: 2 letters are the same, and 3 letters are different.
- Possible combinations: "aaXYZ".
- Choices for A: A, B, C (since they appear at least 2 times).
- If we select A, then X, Y, Z can be chosen from B, C, D, E.
- Total combinations = 3 (for A, B, C) * (number of ways to choose 3 different letters from remaining letters).
- Remaining letters after choosing A: B, C, D, E (4 letters).
- Number of ways to choose 3 different letters from 4 = C(4, 3) = 4.
- Total = 3 * 4 = 12 ways.

**Case 5**: 1 letter is the same, and 4 letters are different.
- Possible combinations: "A, B, C, D, E".
- We can select any 5 letters from A, B, C, D, E.
- Total = 1 way (since we can only select A, B, C, D, E).

3. **Sum All Cases**:
- Total ways = Case 1 + Case 2 + Case 3 + Case 4 + Case 5
- Total ways = 1 + 4 + 18 + 12 + 1 = 36 ways.

### Final Answer:
The total number of ways to select 5 letters from the string "abacabababcdced" is **36**.