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Question: The number of ways in which 4 students can sit in 7 chairs in a row if there is no empty chair betwe...

The number of ways in which 4 students can sit in 7 chairs in a row if there is no empty chair between any two students is
A) 24
B) 28
C) 72
D) 96

Explanation

Solution

In the question, we are given a total number of chairs and a total number of students that is the value of nn and rr is given by 7 and 4 respectively. We need to find the number of ways of sitting arrangements such that no chair remains empty between any 2 students which imply that all the 4 students are sitting on conservative chairs. So, the number of ways can be calculated by n−r+1n - r + 1 and as every student has 4 choices each so n−r+1n - r + 1 is multiplied by 4!4!.

Complete step by step solution: Consider the data given that is the total number of chairs are given as 7 which means n=7n = 7 and the total number of students are given as 3 which means r=3r = 3.
As we have to calculate the number of ways when no chair remains empty between any 2 students, we have to select 4 conservative chairs between the 4 students.
Thus, we get,

n−r+1=7−4+1 =4 n - r + 1 = 7 - 4 + 1 \\\ = 4 \\\

Hence, we know that there are 4 students and the number of possibilities of interchanging the 4 chairs would be factorial of 4.
So, the total number of arrangements of the chairs such that no chair remains empty between 4 students are given by 4×4!4 \times 4! as there are four students and all 4 students have 24 choices that are 4!4! choices to made that is why we will multiply 4 by 4!4!.
Thus, we get,

4×4!=4×(4×3×2×1) =96 4 \times 4! = 4 \times \left( {4 \times 3 \times 2 \times 1} \right) \\\ = 96 \\\

Thus, there are 96 ways in which 4 students can sit in a position on 7 chairs such that no seat remains empty between the 2 students.
Hence, option D is correct.

Note: For calculating several ways we use the formula n−r+1n - r + 1. In questions like this after calculating the value of n−r+1n - r + 1, we need to multiply with factorial rr, several arrangements can only be final when all the ways are multiplied as in this question, we have 4 students so each student can repeat the process which gives us the total number of arrangements by multiplying 4 by 4!4!.