Acard is drawn at random from a deck of fifty-two cards. what is the probability of drawing a diamond, a card with an even number on it, or a card with a number divisible by three on it, but not a card that falls into more than one of these categories

The probability of a diamond is 13/52 = 1/4 So none of the diamonds can be used for the other 2 restrictions.

The even numbered cards are 2 4  8 10.
The 6 cannot be used, because it could come into the next category. Six is divisible by 3. 
These can come from clubs hearts and spades. 
So each even number has 3 suits it can come from
The number of cards in this category is 3 * 4 = 12
The probability of this occurring is 12/39 = 4/13

The cards divisible by 3 are
3 and 9 [Again you can't use the 6]
The Total number of cards are 2*3 = 6
The probability is 6/39 = 2/13

The total probability (without drawing a 6) is
1/4 + 2/13 + 4/13 = 37/52

The two categories where a 6 can turn up are equally likely so choosing the category is 1/2. the probability of a 6 turning up in the first category is 1/2* 3/39 = 3/78 = 1/26

The probability of the 6 turning up in the second category is 1/2 * 3/39 = 3/78 = 1/26 So these two numbers must added into the mix.

37/52 + 2(1/26) = 37/52 + 1/13 = 41/52