Group Selection with Conditions

The set of available numbers is $\{1, 2, 3, 4, 5, 6, 7, 8\}$. 3 and 5 are mandatory. So we must choose at least 1 more element from $\{1, 2, 4, 6, 7, 8\}$. Total elements in remaining set $= 6$. Condition: 7 and 8 are never included together.

Let's count valid choices of remaining elements: Total subsets of $\{1, 2, 4, 6, 7, 8\} = 2^6 = 64$. Subsets containing BOTH 7 and 8: choose 7 and 8 (mandatory), plus any subset of $\{1, 2, 4, 6\} \implies 2^4 = 16$. So subsets NOT containing both 7 and 8 $= 64 - 16 = 48$.

We need groups of size THREE OR MORE. Since 3 and 5 are already included, the subset of remaining elements must have size $\ge 1$. The empty subset (size 0) contributes a group of size 2 (only {3, 5}), which is invalid. The empty subset does not contain both 7 and 8, so it is counted in the 48. Subtracting this 1 invalid empty subset: $48 - 1 = 47$.