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rungtamehul
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We know that,
There are 180 different targets
Archer earns 1 point for target hit and loses 0.33 point for a target missed
No 2 archers shot at same number of targets
Every archer scored 36 points

We need to find out maximum possible no of archers

Let,
Number of targets hit = H
Number of targets missed = M
Total number of targets = T
=> T = H + M

Based on the scoreing system
H - M/3 = 36
=> M = 3H - 108

Substituting this in T = H+M
=> T = H + (3H - 108)
=> T = 4H - 108

Now we know that misses cant be a negative number
=> M >= 0
=> 3H - 108 >= 0
=> H >= 36

=> Minimum number of hits an archer can have is 36, where he gets 36 points from hits and loses 0 points from misses

We also know that total number of targets is 180
=> Maximum number of shots <= 180
=> T <= 180
=> 4H - 108 <= 180
=> H <= 72

Therefore, maximum number of shots archer can hit is 72 where the remaining 108 can be missed to get a total of 36 points ( 72 - 108/3 = 36)

So far we have,
36 <= H <= 72

=> Total archers = 72 - 36 + 1 = 37

Maximum possible number of archers = 37

C. 37
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Max 180 targets
Let’s consider archer shot all of them once
To score 36, he needs to get 36 correct and rest should cancel i.e. for every one correct there will be 3 incorrect. So sets of 4
In remaining 180-36 =144 shots, pairs of 4 are created which means 36 pairs are there so this is the maximum

For second case, lets pick a case below this,
This time to score 36, we need 36 correct and pair of 4 which will cancel out with each other.
Since no archer shot same no, if we consider just one less, there will be 35 pairs.

Similarly for rest if cases go like 34, 33, ...0
I am considering 0 as last case as there could a scenario where cancel out pairs are not shot.


And we cannot go below 0 as we need atleast 36 points

So total possibilities for different number of shots is 37 cases
Hence answer is (C) 37
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