I am struggling with combinations and permutations. One particular concept that is bugging me is selecting outcomes.
I posed a few questions in a forum.
\What is the probability that you are dealt a "full house"? (Three cards of one rank and two cards of another rank.)\
I received following answer
""When counting the number of full house hands you have to choose from which of your two selected ranks the 3 suits will come and from which the 2 suits will come. Two ways to calculate this ${13 \choose 2} {2 \choose 1} {4 \choose 2} {1 \choose 1} {4 \choose 3}$ or ${13 \choose 1} {4 \choose 2} {12 \choose 1} {4 \choose 3}$""
but again when I tried to solve another problem using the same method,
Another Question: What is the probability that you are dealt a "two pair"? (Two pairs of cards where each pair contains two cards of the same denomination, with the fifth card of a different denomination. Note that we exclude four of a kind from this definition.)
My approach is as follows
$\large{\frac{{13 \choose 2} {2 \choose 1}{4 \choose 2}{1 \choose 1}{4 \choose 2}{52-4-4 \choose 1}}{{52 \choose 5}}}$
but it wasn't quiet right,
and when I asked why this does not work, I was told
"In the full house case it was necessary to distinguish which rank had 3 cards and which rank 2 cards. In this case there is no way to distinguish because both ranks have the same number of cards. ${13 \choose 2}{4 \choose 2}{4 \choose 2}{11 \choose 1}{4 \choose 1}$""
This concept is quiet new to me.
Apparently I am missing something basic about combinations. What would be a good book or webpage or any kind of educational material, where I can practice these type of combination and permutation problems? I would also value comment from educators who have seen students struggling to solve these type of counting problems.