A System with $N$ particles and Two states Could you please help me with this question?

Consider a system of $N$ objects that can be in either of two states: up ($U$) or down ($D$). Assume that the probability of the particles being in either state is $0.5$, and that their states are independent of one another. Also assume that all the possible arrangements have equal probabilities. What is the probability of finding a system within a range $45–55\%$ of the objects being in the up state if i) $N = 100$, ii) $N = 10 000$, iii) an arbitrarily large number. 

The assumption that all the possible arrangements have equal probabilities doesn't make sense. I calculated the probability for $N=100$. My answer is $0.73$, however, for $N=10000$ I cannot add all the probabilities. It's a large number. I just can say that the probability is higher than that of the system with $100$ particles. Is that right? 
 A: Hints:  Here are some probabilities from R statistical
software.
If $n = 100,$ then let $X \sim \mathsf{Binom}(100, 0.5)$ and find 
$$P(45 \le X \le 55) = P(X \le 55) - P(X \le 44) = 0.7287,$$
as found using R statistical software, where pbinom is a binomial CDF.
diff(pbinom(c(44,55), 100, .5))
[1] 0.728747

Similarly, for $n = 1000,$
$$P(450 \le X \le 550) = 0.9986.$$
diff(pbinom(c(449,550), 1000, .5))
[1] 0.9986083

Similarly, for $n = 10,000,$ the answer is so close to $1$ that R just prints 1.
diff(pbinom(c(4500-1, 5500), 10000, .5))
[1] 1


By using the normal approximation to binomial with continuity correction (for 'small' values of $n),$ you should be able to get good
approximations of such probabilities, and perhaps find
a formula in $n$ and the standard normal CDF $\Phi().$
Here is an illustration of the best-fitting normal distributions for the two cases you mentioned explicitly.

par(mfrow=c(2,1))

x = 0:100;  PDF = dbinom(x, 100, .5)
plot(x, PDF, type="h", lwd=2, main="BINOM(100,.5) with Approx. Normal Density")
  abline(h = 0, col="green2")
  abline(v = c(44.5,55.5), col="red", lwd=2, lty="dotted")
  curve(dnorm(x, 50, 5), add=T, lwd=2, col="blue")

x = 400:600;  PDF = dbinom(x, 1000, .5)
plot(x, PDF, type="h", main="BINOM(1000,.5) with Approx. Normal Density")
  abline(h = 0, col="green2")
  abline(v = c(449.5,550.5), col="red", lwd=2, lty="dotted")
  curve(dnorm(x, 500, sqrt(250)), add=T, lwd=2, col="blue")

par(mfrow=c(1,1))

