Permutations and Combinations. What are tips and tricks for solving complicated Permutations and Combinations?
1 Answer
See below on some ideas:
Explanation:
Let me answer the original question by answering the comment:
"how many different 5 card hands are possible containing at least 3 black cards in a standard deck of cards? How do I set up a question like this in a simple way."
The answer, at least for me, is to work through the question bit by bit. As the questions become more complicated, you can't simply jump into the middle of it and expect to know the answer. For example, just looking at the question from the comment, I'm not exactly sure what I'll be doing. And so first, I sit with the question for a minute.
We want at least 3 black cards in our hand, and so let's force pick 3 black cards - if more come along for the ride, fine. And so the set up will be to calculate the number of ways we can pick 3 black cards from the 26 available, and then multiply by the number of ways we can pick the remaining 2 cards from the remaining 49 cards in the deck:
And then work out the math.
And so I think for me, the most important things are:
-
Don't jump into the middle of the problem. Whittle away at the question. This will also help you not feel overwhelmed.
-
It takes time to get to the solution on more complicated questions. Take that time.
-
Look for ways to minimize work. We could have done the example above by looking specifically at hands with exactly 3 black cards, then 4, then 5, and added them all up. But we didn't need to, so why do it?
-
A last thing I like to do is to simply be amazed at the answers that come up. For instance, people like to talk about how many stars are in the sky or how many grains of sand are on a beach, but I like to talk small things that create huge numbers - like the nearly 2.6 million number of poker hand combinations. One pack of cards, 2.6 million 5-card hand possibilities. That blows me away.
