Question 1:
I am working on probabilities and on some exercises the solutions either use the nCk or just n.
I want to find a method to understand when I have to use the nCk and when not to.
Example 1 (nCk is used):
What is the probability that a five-card poker hand contains the two
of diamonds, the three of spades, the six of hearts, then ten of
clubs, and the king of hearts?
Solution: 1/ 52 C 5
Example 2 (nCk is not used):
What is the probability that a five-card poker hand does not contain
the queen of hearts?
Solution: 47/52
Two similar questions with different Sample Space!
Is any tip or standard rule that can be used to determine whether a combination or Permutation must be used, and when just the number of all possibles events (e.g S=52) ?
Question 2:
Some exercises do not determine which method must be used (Combinations or Permutations). As I know, Combinations is used when order doesn't matters and repetition is not allowed, on the other hand Permutations is used when order matters and repetition is not allowed.
Both have another kind of method where the repetition is allowed (n+k,C,k and n^k respectively).
Any tip or trick when should I use the Combinations or Permutations? and when to use the unlimited repetition for both? I knew that exercise must make it clear, but I find it a bit hard to deal with this choice of the method.
Thank you in advance.
Best Answer
You can use even permutations in you first question! Why not? It will be just multiplying $5!$ in both numerator and denominator which just get cancelled out!
Also in the second case : $\dfrac{^{51}C_5}{^{52}C_5}=\frac{47}{52}$
First represents choosing $5$ cards from pack without the heart queen and denominator from all cards.
Both give same result.
And generally the question states what to do. If it is unclear, just do what you feel easier. Because people forget to explain elementary cases. But, still, the question should make it clear.