# There is a contest with 40 Pokemons. There are 18 Pokemons who like to ﬁght in the sky, and 23 who like to ﬁght on ground. Several of them like to ﬁght in water. The number of those who like to ﬁght in the sky and on ground is 9. There are 7 Pokemons who like to ﬁght in the sky and in water, and 12 who like to ﬁght on ground and in water. There are 4 Pokemons who like to ﬁght in the sky, on ground, and in water. How many Pokemons like to ﬁght in water How do i express this problem with discrete math?

There is a contest with 40 Pokemons. There are 18 Pokemons who like to ﬁght in the sky, and 23 who like to ﬁght on ground. Several of them like to ﬁght in water. The number of those who like to ﬁght in the sky and on ground is 9. There are 7 Pokemons who like to ﬁght in the sky and in water, and 12 who like to ﬁght on ground and in water. There are 4 Pokemons who like to ﬁght in the sky, on ground, and in water. How many Pokemons like to ﬁght in water. How do i express this problem with discrete math?
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Step 1
$N\left(sky\cup ground\cup water\right)=N\left(sky\right)+N\left(ground\right)+N\left(water\right)-N\left(sky\cap ground\right)-N\left(sky\cap water\right)-N\left(ground\cap water\right)+N\left(sky\cap ground\cap water\right)$
Step 2
where N(X) is number of pokemon that wants to fight in terrain X. This is called inclusion-exclusion principle.
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Step 1
Assuming there every Pokemon wants to fight.
We can use principle of inclusion and exclusion to solve this problem. If x is the number of Pokemons like to ﬁght in water then,
$40=18+23+x-\left(9+7+12\right)+4⇒x=23$
Step 2
If there are Pokemon who don't want to fight then we need to subtract that number from the RHS of the equation and then solve for x.
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