1. **Problem statement:** Twelve people (7 Canadians and 5 Australians) apply for 5 jobs at a ski resort. We want to find the number of ways to award these jobs under different conditions. 2. **General formula:** The number of ways to choose $k$ people from $n$ is given by the combination formula: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ where $n!$ is the factorial of $n$. 3. **Part (a): No restrictions** - We choose any 5 people from 12. - Number of ways: $$\binom{12}{5} = \frac{12!}{5!7!} = 792$$ 4. **Part (b): Only Canadians can be hired** - We choose all 5 from the 7 Canadians. - Number of ways: $$\binom{7}{5} = \frac{7!}{5!2!} = 21$$ 5. **Part (c): Two jobs must go to Canadians** - Choose 2 Canadians from 7: $$\binom{7}{2} = 21$$ - Choose remaining 3 Australians from 5: $$\binom{5}{3} = 10$$ - Total ways: $$21 \times 10 = 210$$ 6. **Part (d): At least two jobs go to Canadians** - This means 2, 3, 4, or 5 Canadians are chosen. - Calculate sum: - 2 Canadians: $$\binom{7}{2} \binom{5}{3} = 21 \times 10 = 210$$ - 3 Canadians: $$\binom{7}{3} \binom{5}{2} = 35 \times 10 = 350$$ - 4 Canadians: $$\binom{7}{4} \binom{5}{1} = 35 \times 5 = 175$$ - 5 Canadians: $$\binom{7}{5} \binom{5}{0} = 21 \times 1 = 21$$ - Total ways: $$210 + 350 + 175 + 21 = 756$$