Question

3 boys and 4 girls have bought tickets for a row of 7 seats at a movie. In how many ways can they arrange themselves in the seats?

Answer 1

If we suppose all the girls and boys are identical, so all that matters is which seats are boys and which are girls, we just have to pick 3 of the 7 seats to put the boys in, or 7 choose 3. This is called the binomial coefficient and the formula for it is\[7!/[3!*(7-3)!]\] (note that we can also choose to pick the 4 seats for the girls first - this is 7 choose 4, but [k choose n] = [k choose (k-n)], so the two are equal).
7 choose 3 = 35.

If we assume each of the 7 people is unique, we simply need to know how many ways there are to arrange 7 objects - 7!
7! = 5040

Answer 2

\[7!=1\times 2\times3\times4\times5\times6\times7=5040\]