Question

There are n different objects 1, 2, 3,......n distributed at random in n places marked 1, 2, 3, ......n. The probability that at least three of the objects occupy places corresponding to their number is

• A. $\frac { 1 } { 6 }$
• B. $\frac { 5 } { 6 }$
• C. $\frac { 1 } { 3 }$
• D. None of these

Answer 1

Answer: (A)

$\frac { 1 } { 6 }$

Answer 2

Let $E _ { i }$ denote the event that the object goes to the place, we have $P \left( E _ { i } \right) = \frac { ( n - 1 ) ! } { n ! } = \frac { 1 } { n } , \forall i$

and $P \left( E _ { i } \cap E _ { j } \cap E _ { k } \right) = \frac { ( n - 3 ) ! } { n ! }$ for $i < j < k$

Since we can choose 3 places out of n in ${ } ^ { n } C _ { 3 }$ ways

The probability of the required event is ${ } ^ { n } C _ { 3 } \cdot \frac { ( n - 3 ) ! } { n ! } = \frac { 1 } { 6 }$.