Answer these questions based on the following information.
A set contains 9 letters such that
S={A,A,B,B,C,C,C,C}.
Find the number of ways of selection of 4 letters such that each of A,B,C must be there.

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem of selecting 4 letters from the set \( S = \{A, A, B, B, C, C, C, C\} \) such that each of A, B, and C must be included, we can follow these steps:

### Step 1: Understand the requirement
We need to select 4 letters from the set, and it is mandatory that we include at least one A, one B, and one C. This means we will start our selection with A, B, and C.

### Step 2: Select A, B, and C
Since we must include one A, one B, and one C, we can denote our current selection as:
- 1 A
- 1 B
- 1 C

This accounts for 3 letters. We now need to select one more letter to complete our selection of 4 letters.

### Step 3: Determine the options for the fourth letter
The fourth letter can be chosen from the remaining letters in the set. After selecting 1 A, 1 B, and 1 C, we have the following letters left:
- 1 A (since we initially had 2 A's)
- 1 B (since we initially had 2 B's)
- 2 C's (since we initially had 4 C's)

Thus, the remaining letters available for selection are:
- A
- B
- C
- C

### Step 4: Count the possible selections for the fourth letter
We have the following options for the fourth letter:
1. Selecting another A (which gives us A, A, B, C)
2. Selecting another B (which gives us A, B, B, C)
3. Selecting a C (which gives us A, B, C, C)

### Step 5: List the combinations
The valid combinations of letters that include at least one A, one B, and one C are:
1. A, A, B, C
2. A, B, B, C
3. A, B, C, C

### Step 6: Conclusion
Thus, we have a total of 3 valid selections of 4 letters that include at least one A, one B, and one C.

The final answer is **3**.