1. **State the problem:** We need to find how many different ways to select 66 numbers from 63 numbers (1 through 63) where order does not matter. 2. **Formula used:** The number of ways to choose $k$ items from $n$ items without regard to order is given by the combination formula: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ where $n!$ is the factorial of $n$. 3. **Apply the formula:** Here, $n=63$ and $k=6$. $$\binom{63}{6} = \frac{63!}{6!(63-6)!} = \frac{63!}{6!57!}$$ 4. **Simplify the factorial expression:** $$\binom{63}{6} = \frac{63 \times 62 \times 61 \times 60 \times 59 \times 58}{6 \times 5 \times 4 \times 3 \times 2 \times 1}$$ 5. **Calculate numerator:** $$63 \times 62 = 3906$$ $$3906 \times 61 = 238266$$ $$238266 \times 60 = 14295960$$ $$14295960 \times 59 = 843861640$$ $$843861640 \times 58 = 48943975120$$ 6. **Calculate denominator:** $$6! = 720$$ 7. **Divide numerator by denominator:** $$\frac{48943975120}{720} = 67977715$$ **Final answer:** There are $67977715$ different possible selections.