In a queue at a movie theatre, five friends — Riya, Sameer, Tanya, Umesh, and Veer — are standing in a line. Riya and Veer want to stand exactly two people apart from each other (i.e., there must be exactly two people standing between them). How many different arrangements of the five friends are possible such that this condition is satisfied?
(a) 12
(b) 18
(c) 24
(d) 36

Question

In a queue at a movie theatre, five friends &mdash; Riya, Sameer, Tanya, Umesh, and Veer &mdash; are standing in a line. Riya and Veer want to stand exactly two people apart from each other (i.e., there must be exactly two people standing between them). How many different arrangements of the five friends are possible such that this condition is satisfied?
(a) 12
(b) 18
(c) 24
(d) 36

ONLiNE UPSC · Accepted Answer

Riya and Veer must be placed in a way that there are exactly two people between them.
This can happen in the following valid positions:
Riya in position 1, Veer in position 4.
Riya in position 2, Veer in position 5.
Veer in position 1, Riya in position 4.
Veer in position 2, Riya in position 5.
Thus, there are 4 possible ways to place Riya and Veer in the positions.
Once Riya and Veer are placed, there are three remaining people: Sameer, Tanya, and Umesh, who need to be arranged in the remaining three positions.
The number of ways to arrange 3 people in 3 positions is 3!, which equals 6.
Thus, the total number of arrangements is:
4 &times; 6 = 24
Hence, option c is correct.

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In a queue at a movie theatre, five friends — Riya, Sameer, Tanya, Umesh, and Veer — are standing in a line. Riya and Veer want to stand exactly two people apart from each other (i.e., there must be exactly two people standing between them). How many different arrangements of the five friends are possible such that this condition is satisfied? (a) 12 (b) 18 (c) 24 (d) 36