Arranging people

Question

4 students and 3 teachers participate in a relay race. Find the number of possible ways to order them if

1. there are no restrictions,
2. the students must be in consecutive positions,
3. the teachers must not all be in consecutive positions,
4. no two teachers are to be in consecutive positions.

Solution

1. $7!=\mathrm{5,040}∎$
2. $4!\times 4!=576∎$
3. $7!-5!\times 3!=4320∎$
4. $4!\times \left(\genfrac{}{}{0px}{}{5}{3}\right)\times 3!=1440∎$