100 Soldiers riddle One of my friends found this riddle.

There are 100 soldiers. 85 lose a left leg, 80 lose a right leg, 75
lose a left arm, 70 lose a right arm. What is the minimum number of
soldiers losing all 4 limbs?

We can't seem to agree on a way to approach this.
Right off the bat I said that:
85 lost a left leg, 80 lost a right leg, 75 lost a left arm, 70 lost a right arm.
100 - 85 = 15
100 - 80 = 20
100 - 75 = 25
100 - 70 = 30
15 + 20 + 25 + 30 = 90
100 - 90 = 10

My friend doesn't agree with my answer as he says not all subsets were taken into consideration. I am unable to defend my answer as this was just the first, and most logical, answer that sprang to mind.
 A: Your answer is right. Think of it like this: How many people are free of not losing each limb? Make sure these people do not overlap, to create a group of people who are guaranteed to have at least one limb and to maximize the size of this group. The remaining people unfortunately do not have this "one limb is safe" guarantee. Calculate their percentage (10%).
A: If you add up all the injuries, there is a total of 310 sustained. That means 100 soldiers lost 3 limbs, with 10 remaining injuries. Therefore, 10 soldiers must have sustained an additional injury, thus losing all 4 limbs.
The manner in which you've argued your answer seems to me, logical, and correct.
A: agree with 10:
minimum number of amputees with both legs missing: ( 100 - (100 - 85) + (100 - 80) ) = 65
minimum number of amputees with both arms missing: ( 100 - (100 - 75) + (100 - 70) ) = 45
now use the same "venn" analysis to find minimum overlap between the previously calculated #s
minimum number of amputees with all limbs missing: (100 - (100 - 65) + (100 - 45) ) = 10
A: 10 and this is why. 


*

*The smallest set of injured soldiers (ones who lost their right arm) were 70 in number. That leaves 30 out of 100 who kept their right arm. Therefore, the largest number of soldiers without the left arm and with the right arm can only be 30 at most. This means that the sets of 70 and 75 must overlap by at least 45.

*We now have a set of 45 soldiers who are missing both arms. That leaves 55 who haven't lost both arms. The largest number of soldiers who have lost their right leg and haven't lost both arms can only be 55 at most. This means that the sets of 45 and 80 must overlap by at least 25.

*We now have a set of 25 soldiers who are missing both arms and their right leg. The sets of 85 and 25 must overlap by at least 10.

A: 80, 85, 70, 75
Let's do this
How many lost both legs?
Well, 80+85= 165. So at least 65 people lost both legs.
How many lost both arms? Well 70+75= 145. So 45 people lost both arms and legs.
Now how many lost all legs and arms?
At the minimum 45 people lost both legs. At the minimum 65 people lost both arms.
So total there are 110 people that lost both arms and legs if no body lost all. Given that there are only 100 people, then we figure that 10 must have lost all.
A: You can easily do it visually with a Venn diagram with the four sets of soliders with each limb. For mimimum number of soliders losing all four limbs, none of the inner sets overlap. So $100 - (15+20+25+30) = 10$.
A: Here is a way of rewriting your original argument that should convince your friend:

Let $A,B,C,D\subset\{1,2,\dots,100\}$ be the four sets, with $|A|=85$,$|B|=80$,$|C|=75$,$|D|=70$.  Then we want the minimum size of $A\cap B\cap C\cap D$.  Combining the fact that $$|A\cap B\cap C\cap D|=100-|A^c\cup B^c\cup C^c\cup D^c|$$ where $A^c$ refers to $A$ complement, along with the fact that for any sets $|X\cup Y|\leq |Y|+|X|$ we see that $$|A\cap B\cap C\cap D|\geq 100-|A^c|-|B^c|-|C^c|-|D^c|=10.$$ 

You can then show this is optimal by taking any choice of $A^c$, $B^c$, $C^c$ and $D^c$ such that any two are disjoint.  (This is possible since the sum of their sizes is $90$ which is strictly less then $100$.) 
A: I agree with your answer.  To have the minimum number lose all four limbs, you want everybody else to lose 3.  You have organized it this way-by adding 15,20,25, and 30 you have argued these sets are distinct.
A: Don't ask how many soliders have lost all limbs at least, but how many have saved a limb, at most.
There are 100 soldiers. 15 saved the left leg, 20 saved the right leg, 25 saved the left arm, 30 saved the right arm.  Together they have only 90 limbs left, so at most 90 can have one.
That means at least 10 have lost all 4 limbs.
