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Suppose we uniformly and randomly select a permutation from the $20 !$ permutations of $1, 2, 3\ldots ,20.$ What is the probability that $2$ appears at an earlier position than any other even number in the selected permutation?

  1. $\left(\dfrac{1}{2} \right)$
  2. $\left(\dfrac{1}{10}\right)$
  3. $\left(\dfrac{9!}{20!}\right)$
  4. None of these
asked in Probability by Veteran (69k points)
edited by | 2.3k views
Hi Guys,

Just small information some people have posted some MadeEasy solution. I would like to say their approach is correct but answer is wrong. If you will solve their equation then also you will get option (B) as answer.

 (1/20)+(1/38)+(1/76)+(2/(19*17))+(7/(19*17*8))+(7/(20*19*17))+(1/(19*17*8))+(1/(26*19*17))+(1/(26*19*17*4))+(1/(26*22*19*17))+(1/(26*22*19*17*10)) = 1/10

But in exam i think this aproach will take more time. So follow the approach as suggested by @Arjun sir.

6 Answers

+30 votes
Best answer
There are $10$ even numbers $(2,4\ldots 20)$ possible as the one in the earliest position and all of these are equally likely. So, the probability of $2$ becoming the earliest is simply $\dfrac{1}{10}$.
answered by Veteran (346k points)
edited by

@Arjun can you please point out why the following solution is wrong, thanks.

@arjun sir

2 appears at an earlier position than any other even numbe?
Their approach is correct but specified answer is wrong.
@Arjun Sir, I didn't quite get the question. What is meant when it says

'2 appears at an earlier position than any other even number in the selected permutation?' What is the position here? Is it the position in the value of a permutation of any no between 1 to 20? Please clarify.

@ shraddha priya
we have numbers from 1 - 20
Question says that their permutation (arrangement) is 20! (that is obvious)
i.e like 1,2,3 .......20
            2,3,4 ......1,20
Like this we have 20! arrangements

Now, 

'2 appears at an earlier position than any other even number in the selected permutation?' 
Means 
That the no. 2 should appear before  all other even no. in that particular arrangement you have selected.

@Arjun  Sir, I am unable to get the solution,

In rosen,symmetry of number has mentioned

2 being earlier in the set of even numbers is 2 is placed in first position , rest all can be in 9!   so it will be 1*9!/10! =1/10 

Is this way correct, [ in any no a,b   a->b is same as b->a ]

+9 votes

Total permutation= 20!

2 come before any other number= first we have to fix position for 10 even number so we can do it in 20c10 ways. Now other than 2 can permuted in 9! ways and 10 odd num can be prmuted in 10! ways

so Ways in which 2 comes before any other even number is-  20C10 *10!89!=20!/10

So probability= 1/10

 

answered by Active (1k points)

Just small correction in place of  20C10 *10!89! it should be  20C10 *10!*9! = 20! / 10

So the prob = (20! / 10) / 20! =1/10

When you are separately doing 10! and 9!, isn't it indicating that the even nos. are separately getting rearranged among themselves and odd nos. are getting rearranged among themselves only. For eg. if 1,2,3,4 is the given set, then it is like (1,2,3,4), (2,1,3,4), (2,1,4,3), (1,2,4,3). It does not include the cases like (2,3,1,4) or (1,4,2,3) and so on, that is the intermixing of odd and even numbers.
+8 votes

 

The odd numbers do not matter here. The probability 2 comes before the other 9 evens is

 

(# of ways to pick 2)(# of ways to pick remaining evens)/(# of ways to order 10 evens)

1*9!/10!=1/10

 

answered by Loyal (4.6k points)
+5 votes
Answer is 1/10 .
(read above made easy solution image attached and read Amitabh tiwari comment )

 Now My solution-

 Made easy solution is correct , if you solve this ( a big calculation) you'll get 1/10 as answer.
So, here calculation is so big so we will solve a small example and try to find out pattern.

Example 1- if i have four numbers 1,2,3,4.  then total permutation is 4! .

Now, same condition is here  " What is the probability that 2 appears at an earlier position than any other even number in the selected permutation?"

now ,  we will solve using method
Number of permutations with 2 in first position    -->   2  _  _  _  =   3!=  6

Number of permutations with 2 in second position    -->   _   2   _   _ =  2 *2!=  4     

Number of permutations with 2 in third position    -->    _  _  2  _  =  2*1*1= 2

so probability is =    6+4+2  / 4!   =  12 /24 = 1/2
    
So Answer is 1/2 .

Second method as Arjun Sir explained -

There are 2 even numbers (2,4) possible as the one in the earliest position and all of these are equally likely. So, the

probability of 2 becoming the earliest is simply 1/2.




Example 2- if i have Six numbers 1,2,3,4,5,6.  then total permutation is 6! .

Now, same condition is here  " What is the probability that 2 appears at an earlier position than any other even number in the selected permutation?"

now ,  we will solve using method
Number of permutations with 2 in first position    -->   2  _  _  _  _  _ =   5!

Number of permutations with 2 in second position    -->   _   2   _   _  _  _ =  3 *4!     

Number of permutations with 2 in third position    -->    _  _  2  _   _   _=  3*2*3!

Number of permutations with 2 in Fourth position    -->    _  _  _  2   _   _= 3*2*1*2!  

so probability is =  5! + 3*4! + 3*2 *3! + 3*2*1*2! / 6!  =  40 / 120 = 1/3
    
So Answer is 1/3 .

Second method as Arjun Sir explained -

There are 3 even numbers (2,4,6) possible as the one in the earliest position and all of these are equally likely. So, the

probability of 2 becoming the earliest is simply 1/3.


Please check ....
answered by Loyal (3.4k points)
+1 vote

This solution is given in Made easy book. please verify :(

answered by (437 points)
madeeasy solution can be verified only by madeeasy.
hifi solution
lol.
This solution is absolutely correct!. Even this will lead to 1/10.

try with 1,2,3,4 using this technique,  you will get it
Their approach is correct but specified answer is wrong.
0 votes

Here order of odd numbers doesn't matter, so we focus only on 10 even numbers as if we have to arrange only 10 even numbers . In general, digit 2 can be placed at any of the 10 places available, but according to question, 2 can be placed at only 1st because it has to appear before every other even number. So out of 10 choices, we have only 1 favourable choice, so probability is 1/10. So option (B) is correct.

answered by (29 points)


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