Problem Solving

coolhabhi wrote:Four different objects 1,2,3,4 are distributed at random in four places marked 1,2,3,4. What is probability that none of the objects occupy the place corresponding to it's number?
A) 17/24
B) 3/8
C) 1/2
D) 5/8

First of all, if we IGNORE the condition about where the objects can be placed, we can arrange the 4 different objects in 4! ways (= 24 ways).

So, we now must determine HOW MANY of those 24 arrangements are such that no objects occupy the location corresponding to its number.
A quick way to do this is to LIST acceptable outcomes.

IMPORTANT: We'll list each arrangement so that the first number represents the object that goes to location #1, the second number represents the object that goes to location #2, and so on.
So, for example, 3421 represents object #3 in location #1, object #4 in location #2, object #2 in location #3, and object #1 in location #4.

Let's be systematic:

Arrangements where object #2 is in location #1
The possible arrangements where NO object is in the correct location are as follows:
2143
2341
2413
Total # of arrangements = 3

Arrangements where object #3 is in location #1
The possible arrangements where NO object is in the correct location are as follows:
3142
3412
3421
Total # of arrangements = 3

Arrangements where object #4 is in location #1
The possible arrangements where NO object is in the correct location are as follows:
4123
4312
4321
Total # of arrangements = 3

Altogether, the number of arrangements where no object is in the correct location = 3 + 3 + 3 = 9

So, P(no object in correct location) = 9/24 = [spoiler]3/8 = B[/spoiler]

Cheers,
Brent

kvcpk wrote:
Looking at the question, it is more obvious that at least 1 number would fall into its corresponding place.
So definitely, the probability that the number wouldn't fall into its numbered place should be less than half.

Looking at options :
17/24
9/24
12/24
15/24

Only option B is less than half. I would pick B and move on. Experts can comment on whether this would be a right approach.

That's a great approach that allows you to minimize time spent on a question (that you feel is going nowhere) and maximize your guess.
Probability questions are perfect for this, because most people have a gut feeling about how likely something is. As you suggest, it does seem unlikely (probability less than 0.5) that every object would be out of place, so B is the perfect guess.

Cheers,
Brent

raj44 wrote:Good Day,

Since there's only one correct way of placing the objects as per the boxes- can't we work on the lines of using probability of an event happening = 1 - probability of that event not happening?

So therefore, can't the answer be 1 - (1/24) = 23/24 ?

Thanks

That's a good idea, but it doesn't apply here.

If event A = NONE of the objects in the correct places, then the complement (event A NOT happening) will consist of any arrangement where some (perhaps all) of the objects are in their correct place(s).

Cheers,
Brent



Cheers,
Brent