How many non-empty subsets of $\{ 1 , 2, 3, 4, 5, 6, 7, 8 \}$ consist entirely of odd numbers?
We consider the subset $\{ 1, 3, 5, 7 \}$ which consists only of the odd integers in the original set. Any subset consisting entirely of odd numbers must be a subset of this particular subset. And, there are $2^4 - 1 = \boxed{15}$ non-empty subsets of this 4-element set, which we can easily see by making the choice of including or not including each element.
15
Level 4
Counting & Probability
Problem: How many non-empty subsets of $\{ 1 , 2, 3, 4, 5, 6, 7, 8 \}$ consist entirely of odd numbers?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the number of non-empty subsets consisting entirely of odd numbers from the set {1, 2, 3, 4, 5, 6, 7, 8}, we first identify the odd numbers within this set. The odd numbers are {1, 3, 5, 7}. 

Next, we count the total number of odd numbers, which is 4. The formula for the number of subsets of a set with n elements is given by 2^n. Therefore, the total number of subsets of the set of odd numbers is 2^4, which equals 16. 

However, this count includes the empty subset, which we do not want since we are interested in non-empty subsets. To find the number of non-empty subsets, we subtract the empty subset from the total count. Thus, we calculate 16 - 1 = 15.

Therefore, the final answer is 15.<|END_OF_LATENT|> The final answer is $15$.