1. **State the problem:** We need to find the number of subsets and the number of proper subsets of the set $\{31, 3, 17, 26, 8, 21\}$. The set has 6 elements. 2. **Formula for number of subsets:** The number of subsets of a set with $n$ elements is given by: $$\text{Number of subsets} = 2^n$$ This is because each element can either be in or out of a subset, giving 2 choices per element. 3. **Calculate number of subsets:** Here, $n=6$, so: $$2^6 = 64$$ 4. **Formula for number of proper subsets:** Proper subsets are all subsets except the set itself. So: $$\text{Number of proper subsets} = 2^n - 1$$ 5. **Calculate number of proper subsets:** $$2^6 - 1 = 64 - 1 = 63$$ 6. **Summary:** - Number of subsets = 64 - Number of proper subsets = 63 This means the original answers given (6 and 6) were incorrect because they did not apply the formula correctly.