Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • If n objects are arranged in a row, then the number of ways of selecting three of these objects so that no two of them are next to each other is   <div class='qna-

If n objects are arranged in a row, then the number of ways of selecting three of these objects so that no two of them are next to each other is 

 

Option: 1

\frac{(n-2)(n-3)(n-4)}{6}


Option: 2

{ }^{n-2} C_3


Option: 3

{ }^{n-3} C_3+{ }^{n-3} C_2


Option: 4

None of these

 


Answers (1)

best_answer

Let a_{0}  be the number of objects to the left of the first object  chosen, a_{1} be the number of object between first and second, the numbers of object between the second and the third is a_{2} and number of object right to the third object is a_{3} .

\\\therefore \quad a_0, a_3 \geq 0 \text{ and } a_1, a_2 \geq 1\\ \\Also, \quad a_0+a_1+a_2+a_3=n-3\: \: \: \: \: \: \: \: \: \: \: ...(i)

  \begin{array}{lc} \because & 1+a_0, 1+a_3 \geq 1 \text { and } a_1, a_2 \geq 1 \\ \\\Rightarrow & \left(1+a_0\right)+a_1+a_2+\left(1+a_3\right)=n-1 \end{array}

\therefore Required number of ways =

 \begin{aligned} & ={ }^{n-1-1} C_{4-1}={ }^{n-2} C_3 \\ \\& =\frac{(n-2)(n-3)(n-4)}{6} \end{aligned}

Also,       { }^{n-2} C_3={ }^{n-3} C_3+{ }^{n-3} C_2

Posted by

Rishi

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2025: NIT Rourkela cut-offs for CSE, chemical engineering drop; last years' closing ranks
anam-khan

JEE Main 2025 Live: NTA exam city slip at jeemain.nta.nic.in soon; session 1 schedule
anam-khan

JEE Main 2025 schedule out for session 1; paper 1 from January 22

Latest Question