In this question, the problem is to find the amount of four-digit numbers that have the following characteristics:
All digits are unique.
Does not contain the digits 3 and/or 4.
The number is divisible by 3.
I was going to use inclusion/exclusion for this, which states:
$$N(A\cap{B})=N(A)+N(B)-N(A\cup{B})$$
Take $A$ to be the first two properties, and $B$ to be the third. It must be a $4$ digit number, and cannot contain $3$ or $4$:
$$7\cdot{7}\cdot{6}\cdot{5}$$
Now consider the number of $4$ digit numbers divisible by $3$. The amount of numbers divisible by $3$ between $1$ and $10000$ is $3333$. The amount of numbers divisible by $3$ between $1$ and $1000$ is $333$. Subtract one from the other, and the amount of numbers divisible by $3$ between $1000$ and $10000$ is $3000$.
$$N(A)=7\cdot7\cdot6\cdot5\\ N(B)=3000\\ N(A\cup{B})=9000$$
This comes out to a negative number, which tells me I'm setting up my cases incorrectly. Can anyone tell me what I'm doing wrong?