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

Options

A. `51`
B. `91`
C. `71`
D. `42`

Answer

To solve the problem of selecting 5 letters from the string "abacabababcdced", we need to consider the frequency of each letter in the string and the different cases of selection.

### Step-by-Step Solution:

1. **Count the Frequency of Each Letter:**
- From the string "abacabababcdced", we have:
- A: 5
- B: 4
- C: 3
- D: 2
- E: 1

2. **Case 1: All 5 letters are the same.**
- The only letter that appears 5 times is A.
- So, there is **1 way** to select 5 A's.

**Hint:** Check the frequency of letters to see if any letter appears 5 times.

3. **Case 2: 4 letters are the same, and 1 letter is different.**
- We can choose either 4 A's or 4 B's.
- If we choose 4 A's, we can select 1 letter from the remaining letters (B, C, D, E). There are 4 options.
- If we choose 4 B's, we can select 1 letter from (A, C, D, E). There are also 4 options.
- Total ways = (4 options from A) + (4 options from B) = **8 ways**.

**Hint:** For each selection of 4 same letters, count the different letters that can be chosen.

4. **Case 3: 3 letters are the same, and 2 letters are different.**
- We can choose 3 A's, 3 B's, or 3 C's.
- If we choose 3 A's, we can select 2 different letters from (B, C, D, E). The number of ways to choose 2 from 4 is \( \binom{4}{2} = 6 \).
- If we choose 3 B's, we can select 2 different letters from (A, C, D, E). Again, \( \binom{4}{2} = 6 \).
- If we choose 3 C's, we can select 2 different letters from (A, B, D, E). Again, \( \binom{4}{2} = 6 \).
- Total ways = 6 (from A) + 6 (from B) + 6 (from C) = **18 ways**.

**Hint:** Use combinations to select different letters after choosing the same ones.

5. **Case 4: 2 letters are the same, and 3 letters are different.**
- We can choose 2 A's, 2 B's, or 2 C's.
- If we choose 2 A's, we can select 3 different letters from (B, C, D, E). The number of ways to choose 3 from 4 is \( \binom{4}{3} = 4 \).
- If we choose 2 B's, we can select 3 different letters from (A, C, D, E). Again, \( \binom{4}{3} = 4 \).
- If we choose 2 C's, we can select 3 different letters from (A, B, D, E). Again, \( \binom{4}{3} = 4 \).
- Total ways = 4 (from A) + 4 (from B) + 4 (from C) = **12 ways**.

**Hint:** Remember to select different letters after choosing the same ones.

6. **Case 5: All 5 letters are different.**
- We can select from A, B, C, D, E. Since we have 5 different letters, there is only **1 way** to select all different letters.

**Hint:** Count the total number of unique letters available.

7. **Total Number of Ways:**
- Adding all the cases together:
- Case 1: 1 way
- Case 2: 8 ways
- Case 3: 18 ways
- Case 4: 12 ways
- Case 5: 1 way
- Total = 1 + 8 + 18 + 12 + 1 = **40 ways**.

### Final Answer:
The total number of ways to select 5 letters from the string "abacabababcdced" is **40**.