Finding cardinality/possible combinations of sets Problems:

Let $S = \{1, 2, 3, 4, 5, 6\}$.
(a) How many subsets are there of cardinality $4$? ($15$)
(b) How many subsets of cardinality $4$ have $\{2, 3, 5\}$ as a subset? ($\binom{3}{1} = 3$, but why $\binom{3}{1}$?)
(c) How many subsets of cardinality $4$ contain at least one odd number? ($\binom{6}{4} = 15$ but why does each subset contain at least one odd?)
(d) How many subsets of cardinality $4$ contain exactly one even number? ($\binom{3}{1}$, but why is this also $3$?)

I had no idea how to phrase the topic since I don't really understand what this is. I am having a difficult time with discrete mathematics. I understand how to get the answer to (a) by using pascal's triangle, but don't understand how to go about the other questions. I am at a loss right now, and I have an exam coming up quickly. Thanks for the help in advance.
 A: 
How many subsets have cardinality $4$?

Since the set $S = \{1, 2, 3, 4, 5, 6\}$ has six elements, a four-element subset that contains the three-element subset $\{2, 3, 5\}$ must contain one element of the subset $\{1, 4, 6\}$.  There are $$\binom{3}{1} = 3$$ ways of choosing one of those three elements.  As TMM stated in the comments, the three resulting subsets are 
$$\{1, 2, 3, 5\}, \{2, 3, 4, 5\}, \{2, 3, 5, 6\}$$

How many subsets of cardinality $4$ contain at least one odd number?

Since the set $S$ contains three odd numbers and three even numbers, there can be at most three even numbers in a subset of size $4$.  Therefore, at least one element in any four-element subset of $S$ must be odd.  Since there are 
$$\binom{6}{4} = 15$$
four-element subsets of $S$, there are $15$ subsets of cardinality $4$ that contain at least one odd number.

How many subsets of cardinality $4$ contain exactly one even number?

If a subset of cardinality $4$ contains exactly one even number, then it must contain all three odd numbers and one of the three even numbers in set $S$. There are 
$$\binom{3}{1} = 3$$
ways of choosing one of the three even numbers in set $S$ to be the only even number in the subset of cardinality $4$.  The three subsets that contain exactly one even number are 
$$\{1, 2, 3, 5\}, \{1, 3, 4, 5\}, \{1, 3, 5, 6\}$$
