I How Many Permutations Exist Where M Equals K on Cards?

[WWGD]

Hello,

I have been trying to solve this problem but I can't seem to find a way.

Given are $n$ cards and each card can show one of two values: M or K.

How many possible permutations are there in which there are as many cards with M as there are with K? Given that $n$ is an even amount of cards.

Is it possible to derive a formula for this as a function of $n$? How does one deduce this?

This way makes sense to me ( a rephrasing of omeone else's answer, I think Dr Claude's ) : Assume you need to go from point A to point B along a grid system , where you must go ,say, north (M) j times and east(K) j times in order to arrive at B, i.e. n=2j. How many ways can you do this trip? Once the j places where you make a turn east(north) fully determine the rest of the trip.
This gives you a way of counting paths where M=K. The total number of paths is straightforward.

 

[JohnnyGui]

This is pretty far astray from the original post that this thread is under. I.e. your original question was asked and answered. Some follow-ups, also asked and answered. Now you have a question about inference -- this requires a new thread, at a minimum. Your line of thinking here doesn't make sense to me. With a large enough number of tosses we should be able to estimate probability of heads up to any amount of (in the real world, reasonable) precision. There are a lot of different approaches, and ways to frame the inference problem.

Personally, I think you need to study probability theory first, then revisit these questions in a few months.

I was talking about when those large number of tosses are divided into small number of tosses, each being a trial, and how one can interpret these small trials to deduce the individual chance.
It was my intention to ask the question in my OP as a base that leads to this question. Making a new thread for every question that I have regarding this would seem ineffective to me. I got it eventually figured out though, so nevermind.

This way makes sense to me ( a rephrasing of omeone else's answer, I think Dr Claude's ) : Assume you need to go from point A to point B along a grid system , where you must go ,say, north (M) j times and east(K) j times in order to arrive at B, i.e. n=2j. How many ways can you do this trip? Once the j places where you make a turn east(north) fully determine the rest of the trip.
This gives you a way of counting paths where M=K. The total number of paths is straightforward.

This is a creative way to think about it. It helped me deduce the same formula again. Thanks!