Logo
Jeeify
MCQEasyJEE 2023Applications of P&C

JEE Mathematics 2023 Question with Solution

In an examination, 55 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sit on the allotted seat, is:

  • A

    4444

  • B

    120120

  • C

    6060

  • D

    4848

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: There are 55 students and 55 allotted seats. Find: The number of arrangements in which no student sits on the allotted seat.

This is a problem of derangements, where no object appears in its original position. The formula for the number of derangements of nn objects is

Dn=n!(1−11!+12!−⋯+(−1)n1n!)D_n = n! \left( 1 - \frac{1}{1!} + \frac{1}{2!} - \cdots + (-1)^n \frac{1}{n!} \right)

For n=5n = 5,

D5=5!(1−11!+12!−13!+14!−15!)D_5 = 5! \left( 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} \right)

Now simplify:

D5=120(1−1+12−16+124−1120)D_5 = 120 \left( 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} - \frac{1}{120} \right) D5=120×44120=44D_5 = 120 \times \frac{44}{120} = 44

Therefore, the number of ways is 4444. The correct option is A.

Using Derangement Formula

Given: None of the 55 students should occupy the seat corresponding to the original roll-number arrangement. Find: Total such permutations.

A derangement counts permutations with no fixed point. So we directly use the derangement count for 55 objects.

D5=5!(1−11!+12!−13!+14!−15!)=120(1−1+12−16+124−1120)=120×44120=44\begin{aligned} D_5 &= 5! \left( 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} \right) \\ &= 120 \left( 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} - \frac{1}{120} \right) \\ &= 120 \times \frac{44}{120} \\ &= 44 \end{aligned}

Hence, the required number of arrangements is 4444.

Common mistakes

  • Using 5!=1205! = 120 as the answer is incorrect because 120120 counts all seat arrangements, including those where one or more students sit on their allotted seats. Use derangements, not total permutations.

  • Confusing this with a circular permutation is wrong because the seats are already fixed and allotted by roll numbers. Treat it as a permutation with restricted positions.

  • Applying the derangement formula incorrectly by stopping the alternating series too early gives a wrong value. Include all terms up to 15!\frac{1}{5!} for D5D_5.

Practice more Applications of P&C questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trialMore on Applications of P&C

Related questions