Question

From a group of 10 persons, in how many ways can a selection of 4 persons to be made such that a particular person is always included?

• A. 40
• B. 120
• C. 84
• D. 210
• E. 60

Hint:

"Always included" $\implies$ Subtract from both $n$ and $r$.
"Always excluded" $\implies$ Subtract only from $n$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
When a specific item or person must always be included in a selection, that person's place is fixed. This effectively reduces both the total number of items to choose from and the number of spots remaining to be filled.

Step 2: Key Formula or Approach:
If 1 particular person is always included, the number of ways to choose $r$ people from $n$ is: \[ ^{n-1}C_{r-1} \]

Step 3: Detailed Explanation:
1. Total number of persons ($n$) = 10. 2. Selection size ($r$) = 4. 3. Since 1 particular person is always included, we have already picked 1 person. 4. Remaining persons to choose from = $10 - 1 = 9$. 5. Remaining spots to fill = $4 - 1 = 3$. 6. Number of ways = $^9C_3$. \[ ^9C_3 = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} \] \[ = 3 \times 4 \times 7 = 84. \]

Step 4: Final Answer:
The number of ways to make the selection is 84.