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?
Answers
Let me do it in reverse order!
First I find out the no. of chances in which at least one will obtain the place corresponding to its number, and then subtract it from the whole to get the answer.
First let's find the total outcomes.
The four different objects can be arranged in four places in 4P4 ways, i. e., 4! = 24 ways.
(4P4 means 4! ÷ (4 - 4)! = 4! ÷ 0! = 24 ÷ 1 = 24)
Now find the no. of chances in which objects obtain corresponding places. Assume that the places are arranged in numerical order.
No. of chances in which object 1 only obtain place 1
= 1 × 3! - (1 + 3) = 6 - 4 = 2
(1 × 3! are the no. of arrangements 1, 2, 3, 4 are arranged by 1 only in first place. 1 + 3 is subtracted as 2, 3, 4 obtain corresponding places 1 times together and 3 times by each in 1 × 3! arrangements. )
No. of chances in which object 1 only obtain place 1 is equal to no. of chances in which either 2, 3 or 4 only obtain corresponding places.
No. of chances in which object 2 only obtain place 2 = 2
No. of chances in which object 3 only obtain place 3 = 2
No. of chances in which object 4 only obtain place 4 = 2
Now,
No. of chances in which objects 1, 2 only obtain places 1, 2 respectively
= 1 × 1 × 2! - 1 = 2 - 1 = 1
(1 × 1 × 2! are arrangements 1, 2, 3, 4 are arranged by 1, 2 only in first and second places respectively. 1 is subtracted as 3, 4 obtain corresponding places together 1 times in 1 × 1 × 2! arrangements. )
No. of chances in which objects 1, 2 only obtain places 1, 2 respectively is equal to no. of chances in which the below following cases are occurred.
No. of chances in which objects 1, 3 only obtain places 1, 3 respectively = 1
No. of chances in which objects 1, 4 only obtain places 1, 4 respectively = 1
No. of chances in which objects 2, 3 only obtain places 2, 3 respectively = 1
No. of chances in which objects 2, 4 only obtain places 2, 4 respectively = 1
No. of chances in which objects 3, 4 only obtain places 3, 4 respectively = 1
Now,
No. of chances in which objects 1, 2, 3 only obtain places 1, 2, 3 respectively
= This can't be found out because when 1, 2, 3 are arranged in corresponding places, naturally 4 will come in place 4 !
There is 1 chance in which 1, 2, 3, 4 obtain corresponding places.
Now, add all these 'unfavourable' outcomes! (The figures which are bold!)
2 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 15
∴ Among 24 arrangements, there are 15 arrangements in which at least 1 object obtains its corresponding place.
Subtract this from the total, i. e., subtract 15 from 24.
24 - 15 = 9
∴ Among 24 arrangements, there are 9 arrangements in which none of the objects occupy its corresponding places.
Now find the probability.
Probability = 9 / 24 = 3 / 8
∴ 3 / 8 is the answer.
The total 24 arrangements are given below. Assume places are in numerical order.
1 2 3 4, 1 2 4 3, 1 3 2 4, 1 3 4 2, 1 4 2 3, 1 4 3 2,
2 1 3 4, 2 1 4 3, 2 3 1 4, 2 3 4 1, 2 4 1 3, 2 4 3 1,
3 1 2 4, 3 1 4 2, 3 2 1 4, 3 2 4 1, 3 4 1 2, 3 4 2 1,
4 1 2 3, 4 1 3 2, 4 2 1 3, 4 2 3 1, 4 3 1 2, 4 3 2 1
And 9 favourable arrangements are also given below. Assume places are......uh?!
2 1 4 3, 2 3 4 1, 2 4 1 3,
3 1 4 2, 3 4 1 2, 3 4 2 1,
4 1 2 3, 4 3 1 2, 4 3 2 1
Hope this article may be helpful.
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