To fill up 12 vacancies, there are 25 candidates of which 5 are from SC. If 3 of these vacancies are reserved for SC candidates while the remaining are open to all then the number of ways in which the selection can be made is

Options

A. `""^(5) C_(3) xx ""^(15)C_(9)`
B. `""^(5)C_(3) xx""^(22)C_(9)`
C. `""^(5)C_(3) xx ""^(20)C_(9)`
D. None of these

Answer

To solve the problem of filling 12 vacancies with the given conditions, we can break down the solution into clear steps:

### Step 1: Understand the total vacancies and candidates
We have a total of 12 vacancies to fill from 25 candidates, out of which 5 are from the Scheduled Cast (SC) category.

### Step 2: Identify reserved vacancies
Out of the 12 vacancies, 3 are reserved for SC candidates. This means that we need to fill these 3 vacancies specifically with SC candidates.

### Step 3: Calculate remaining vacancies
After filling the 3 reserved vacancies for SC candidates, the remaining vacancies will be:
\[ 12 - 3 = 9 \]
These 9 vacancies are open to all candidates.

### Step 4: Choose SC candidates
We need to choose 3 SC candidates from the 5 available SC candidates. The number of ways to choose 3 candidates from 5 is given by the combination formula:
\[ \text{Number of ways to choose SC candidates} = \binom{5}{3} \]

### Step 5: Choose remaining candidates
After selecting 3 SC candidates, we have 9 vacancies left to fill. The remaining candidates available for selection will be:
- Total candidates = 25
- SC candidates chosen = 3
- Remaining candidates = 25 - 3 = 22

However, we need to consider that there are still 2 SC candidates left (5 total - 3 chosen), and 20 non-SC candidates. Therefore, we can choose from these 22 candidates (2 SC + 20 non-SC) to fill the remaining 9 vacancies.

The number of ways to choose 9 candidates from these 22 is given by:
\[ \text{Number of ways to choose remaining candidates} = \binom{22}{9} \]

### Step 6: Calculate total combinations
The total number of ways to fill the vacancies can be calculated by multiplying the combinations from Step 4 and Step 5:
\[ \text{Total ways} = \binom{5}{3} \times \binom{22}{9} \]

### Step 7: Compute the values
Now we can compute the values:
1. Calculate \( \binom{5}{3} \):
\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

2. Calculate \( \binom{22}{9} \):
\[ \binom{22}{9} = \frac{22!}{9!(22-9)!} = \frac{22!}{9! \times 13!} \]
(This value can be computed using a calculator or software for large factorials.)

### Final Calculation
Finally, multiply the results from Step 6:
\[ \text{Total ways} = 10 \times \binom{22}{9} \]

### Conclusion
The total number of ways in which the selection can be made is \( 10 \times \binom{22}{9} \).

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