JEE Advance - Mathematics (2018 - Paper 1 Offline - No. 17)
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
(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
$${3 \over {40}}$$
$${1 \over 8}$$
$${7 \over 40}$$
$${1 \over 5}$$
Explanation
Here, five students S1, S1, S3, S4 and S5 and five seats R1, R2, R3, R4 and R5
$$ \therefore $$ Total number of arrangement of sitting five students is 5! = 120
Here, S1 gets previously allotted seat R1
$$ \therefore $$ S2, S3, S4 and S5 not get previously seats.
Total number of way S2, S3, S4 and S5 not get previously seats is
$$4!\left( {1 - {1 \over {1!}} + {1 \over {2!}} - {1 \over {3!}} + {1 \over {4!}}} \right)$$
$$ = 24\left( {1 - 1 + {1 \over 2} - {1 \over 6} + {1 \over {24}}} \right)$$
$$ = 24\left( {{{12 - 4 + 1} \over {24}}} \right) = 9$$
$$ \therefore $$ Required probability = $${9 \over {120}} = {3 \over {40}}$$
$$ \therefore $$ Total number of arrangement of sitting five students is 5! = 120
Here, S1 gets previously allotted seat R1
$$ \therefore $$ S2, S3, S4 and S5 not get previously seats.
Total number of way S2, S3, S4 and S5 not get previously seats is
$$4!\left( {1 - {1 \over {1!}} + {1 \over {2!}} - {1 \over {3!}} + {1 \over {4!}}} \right)$$
$$ = 24\left( {1 - 1 + {1 \over 2} - {1 \over 6} + {1 \over {24}}} \right)$$
$$ = 24\left( {{{12 - 4 + 1} \over {24}}} \right) = 9$$
$$ \therefore $$ Required probability = $${9 \over {120}} = {3 \over {40}}$$
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