On Planet X, cards can take on a numerical value from $1$ to $7$ (inclusive) and their suit can be either red or blue. In a game of poker, each player gets three cards.
1) What is the probability of a straight (e.g. three consecutive cards)? Note that wrap-arounds count, meaning that $7, 1, 2$ would count as a straight
2) What is the probability of getting a flush that is NOT a straight?
3) What is the probability of getting a straight flush?
4) What is the probability of getting a pair? Note that a pair cannot also be a straight or a flush.
5) What is the probability of getting a hand that is not a straight, nor a flush, nor does it contain $2$ of a kind?
I am sort of new to probability and combinatorics so I wanted someone to help me verify these problems. Here are my attempts:
$1$) There are $7 \cdot 2 = 14$ total cards in the deck, so there are ${14\choose 3}$ possible hands. Then I need to count the number of ways to get a straight. I sort of just tried to list them all out... but then it gets complicated since there are two different suits.
I would appreciate it if someone can please help me with this problem. I know that the denominator will be ${14\choose 3}$, but I don't have a clever way to enumerate the numerator.
Also, I think the answer to $5$ will just be the ${14\choose 3}$ minus the sum of everything we've done so far all over ${14\choose 3}$, but I'm not sure about that again.