Question

In a group of \(5\) students, how many ways can the positions of president}, vice-president}, and treasurer} be filled?

• A. \(60\)
• B. \(120\)
• C. \(30\)
• D. \(210\)

Hint:

If posts are distinct}, use permutations (\(^{n}P_{r}\)). If you’re only forming a committee with identical roles, use combinations (\(^{n}C_{r}\)).

Answer 1

The Correct Option is A

Solution:
Step 1 (Identify the counting model).
The three posts are distinct}. One student cannot hold more than one post. Therefore, the arrangement is a permutation of \(3\) students chosen from \(5\).
Step 2 (Count sequentially).
Choose President in \(5\) ways.
Then Vice-President in \(4\) ways (one student already use(d).
Then Treasurer in \(3\) ways.
By the multiplication principle: \[ \text{Ways} = 5 \times 4 \times 3 = 60. \] Step 3 (Permutation formula cross-check).
Using \(^{n}P_{r}=\dfrac{n!}{(n-r)!}\): \[ {}^{5}P_{3}=\frac{5!}{(5-3)!}=\frac{120}{2}=60 \ \ \text{matches Step 2.} \] \[ {60 \ \text{(Option (a)}} \]