Question

Four different objects 1,2,3,4 are distributed at random in four places marked 1,2,3,4. What is the probability that none of the objects occupy the place corresponding to its number ?

• A. $\frac{17}{24}$
• B. $\frac{3}{8}$
• C. $\frac{1}{2}$
• D. $\frac{5}{8}$

Answer 1

The correct option is C $\frac{1}{2}$

Let four places be
1 2 3 4
Now object i cannot occupy the place 1. ...(A)
Suppose object 2 occupies the place 1. Then other placements can be done in following 6 ways :
(i) 2 1 3 4
(ii) 2 1 4 3
(iii) 2 3 1 4
(iv) 2 3 4 1
(v) 2 4 1 3
(vi) 2 4 3 1
Here out of six ways, only there are permissible, because (i), (iii) and (vi) are not permissible because of the non-fulfilment of condition (A).
Hence, required probability is $\frac{3}{6}=\frac{1}{2}$.
Similarly we can allow objects 3 and 4 to occupy place 1 and in each case we can find that the Probability is $\frac{1}{2}$