Quants Questions for CLAT | QB Set 20 [Permutation Problems]

A university is organising a national legal awareness competition in which students from different colleges will participate. The organising committee has selected 8 students from different regions to take part in various activities. Since the opening ceremony requires a special arrangement, the committee needs to decide the order in which five selected students will deliver short speeches. The order of speakers is important because each student represents a different theme of legal awareness, and no two students can occupy the same position.

Apart from speeches, the committee also needs to arrange award certificates for the top performers. There are 6 different certificates that must be placed in a row on the stage table. The placement order of certificates will determine the sequence in which winners are called. The organisers also plan to form a team of 3 students from the selected group for a special presentation, where the order of selection will decide the speaking roles assigned to each member.

Using the concepts of permutation, the committee wants to calculate the possible arrangements for different events and ensure that every arrangement follows the required rules.

Questions

Q1. In how many different ways can 5 students be arranged for delivering speeches if all 5 positions are different?
A) 60
B) 100
C) 240
D) 120

Q2. The committee has 6 different certificates to place in a row. In how many ways can the certificates be arranged?
A) 720
B) 360
C) 120
D) 60

Q3. From the 8 selected students, the committee wants to choose and arrange 3 students for the special presentation. How many different arrangements are possible?
A) 56
B) 336
C) 168
D) 24

Q4. If 4 students are selected from a group of 7 students and arranged for four different speaking roles, how many arrangements are possible?
A) 840
B) 210
C) 120
D) 504

Q5. A team has 5 students, and the committee wants to select a president and a vice-president from them. In how many ways can these two positions be assigned?
A) 5
B) 20
C) 10
D) 25

Answers with Detailed Explanation

1. Answer: D) 120
   The arrangement of 5 students in 5 different positions is a permutation problem.
   Formula: nPr = n! / (n − r)!
   Here, n = 5, r = 5
   5P5 = 5! / (5 − 5)! = 5! / 0! = 5! (since 0! = 1)
   5! = 5 × 4 × 3 × 2 × 1 = 120
   Therefore, the total number of possible speech arrangements is 120.
2. Answer: A) 720
   The 6 certificates are different, and all have to be arranged in a row.
   Number of arrangements = 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
   Therefore, the certificates can be arranged in 720 different ways.
3. Answer: C) 336
   Here, 3 students are selected from 8 students and their order matters because different speaking roles are assigned.
   Therefore, we use permutation:
   8P3 = 8! / (8 − 3)! = 8! / 5! = 8 × 7 × 6 = 336
   Therefore, 336 arrangements are possible.
4. Answer: A) 840
   Here, 4 students are selected and arranged from a group of 7 students.
   Since the order of speaking roles matters:
   7P4 = 7! / (7 − 4)! = 7! / 3! = 7 × 6 × 5 × 4 = 840
   Therefore, the total number of arrangements is 840.
5. Answer: B) 20
   There are 5 students, and two different positions (President and Vice-President) must be assigned.
   Since the positions are different, order matters.
   Using permutation:
   5P2 = 5! / (5 − 2)! = 5 × 4 = 20
   Therefore, the president and vice-president positions can be assigned in 20 different ways.