I am reading over some of my notes and I am having a little trouble understanding combinations and permutations and when to use them.
Problem: How many thirteen-card hands can be chosen from a pack of cards?
My Solution:The order of cards once in the hand does not matter so am I right by saying there are P(52,13)/P(13,13) ways. I.e. C(52, 13) ways?
Problem:On the hands in the previous problem, how many consist of seven cards in one suit and six in another?
- Pick a suit..... 4 ways
- Choose seven cards from this suit, with repetition forbidden and order not important .... C(13,7) ways
- Pick another suit .... 3 ways
- Choose six cards from this suit ... C(13,6) ways
So number of hands is 4*C(13,7)*3*C(13,6) => P(4,2)C(13,7)C(13,6)
This is the solution in my notes but I don't understand in step for why there is C(13,6) ways. Shouldn't there only be C(6,6) ways since we have already picked the first 7 cards?
Problem: What is the number of 12 card hands containing four cards in each of three suits?
Sample Solution: P(4,3)C(12,4)C(12,4)C(12,4) = P(4,3) C(12,4)^3
Again, with this solution why isn't it P(4,3)C(12,4)C(8,4)C(4,4)
Also shouldn't it be C(4,3) not P(4,3) since the order we pick the suits doesn't matter. Ie. 4 Clubs, 4 Spades, 4 Diamonds is the same as 4 Diamonds, 4 Spades, 4 Clubs.
Help would be much appreciated. D: