When 8 coins are tossed simultaneously, in _______ number of the outcomes at most two of the coins will turn up as heads.

To solve this problem, we need to use the concept of combinations in probability.

1. When no coin turns up as heads: There is only 1 way this can happen, which is when all coins turn up tails.

2. When exactly one coin turns up as heads: This can happen in 8 ways. This is because there are 8 coins, and any one of them can turn up as heads.

3. When exactly two coins turn up as heads: This can happen in "8 choose 2" ways. In mathematics, "n choose r" is a way of selecting r items from a larger set of n items. In this case, "8 choose 2" equals 28.

So, the total number of outcomes in which at most two of the coins will turn up as heads is 1 (no heads) + 8 (one head) + 28 (two heads) = 37.