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Probability question

2018 · Shift 1 · Q35
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Probability question

2018 · Shift 1 · Q35

JEE AdvancedMathematicsProbabilityMCQ+3 / −1
There are five students S1, S2, S3, S4 and S5 in a music class and for them there are five seats R1, R2, R3, R4 and R5 arranged in a row, where initially the seat Ri is allotted to the student Si, i = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats. (There are two questions based on Paragraph "A", the question given below is one of them) The probability that, on the examination day, the student S1 gets the previously allotted seat R1, and NONE of the remaining students gets the seat previously allotted to him/her is
  1. A
    340{3 \over {40}}403​
  2. B
    18{1 \over 8}81​
  3. C
    740{7 \over 40}407​
  4. D
    15{1 \over 5}51​
View written solutionFree

Correct answer: A

  1. Interpret the condition

We have 5 students and 5 seats. On the exam day, seats are allotted randomly, so all 5!5!5! permutations are equally likely.

We want the probability that:

  • S1S_1S1​ gets seat R1R_1R1​, and
  • none of S2,S3,S4,S5S_2,S_3,S_4,S_5S2​,S3​,S4​,S5​ gets his/her original seat.

So after fixing S1→R1S_1 \to R_1S1​→R1​, the remaining 4 students must be arranged among R2,R3,R4,R5R_2,R_3,R_4,R_5R2​,R3​,R4​,R5​ in such a way that no one gets the corresponding original seat.

This is exactly a derangement of 4 objects.

  1. Total number of possible seatings

The total number of random seat allocations is

5!=120.5! = 120.5!=120.
  1. Favourable arrangements

First, fix S1S_1S1​ in R1R_1R1​.

Now we need to arrange S2,S3,S4,S5S_2,S_3,S_4,S_5S2​,S3​,S4​,S5​ in seats R2,R3,R4,R5R_2,R_3,R_4,R_5R2​,R3​,R4​,R5​ with no fixed point.

The number of derangements of 4 objects is

!4=9.!4 = 9.!4=9.

So favourable outcomes = 999.

  1. Required probability

Therefore,

P=95!=9120=340.P = \frac{9}{5!} = \frac{9}{120} = \frac{3}{40}.P=5!9​=1209​=403​.
  1. Check options
340\frac{3}{40}403​

is Option A.

  1. Comparison with stored answer

Stored correct answer: A

Our derived answer: A

So they agree.

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