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?
@Arjun can you please point out why the following solution is wrong, thanks.
@ 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
Like this we have 20! arrangements
'2 appears at an earlier position than any other even number in the selected permutation?'
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 ]
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
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
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)
This solution is given in Made easy book. please verify :(
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.
If "Rounded to the nearest integer" is the ...