### Instruction: 8 coins are simultaneously flipped.  What is the probability that heads are showing on at most 2 of them?### Response: I want to find the probability of an event that has several possible outcomes: either 0, 1, or 2 heads out of 8 flips.
To do this, I can use the formula for the binomial probability distribution, which gives the probability of getting exactly k successes out of n trials, each with probability p of success.
In this case, a success is getting a head, so p = 0.5, and n = 8.
The formula is:

P(k) = (n choose k) * p^k * (1-p)^(n-k)
This means I have to calculate the binomial coefficients (n choose k), which are the number of ways to arrange k objects out of n, or n!/(k!(n-k)!), where ! means factorial.
For k = 0, 1, or 2, I get:

(n choose 0) = 8!/(0!8!) = 1
(n choose 1) = 8!/(1!7!) = 8
(n choose 2) = 8!/(2!6!) = 28