How many ways are there to seat six different boys and six different girls along one side of a long table with 12 seats? how many ways if boys and girls alternate seats?

Q1: 479 001 600 ways

Explanation:
You have 12 people, 12 seats, ¹²P₁₂ ways in arranging them, or essentially 12! ways

Q2: 1 036 800 ways

Explanation:
Let's break it up into two cases.

Case 1: BGBGBGBGBGBG
Case 2:GBGBGBGBGBGB

Let's deal with case 1, because case 2 will pop out. There are ⁶P₆ ways in sorting out the boys and ⁶P₆ ways in sorting out the girls, so for case 1, we'd have (⁶P₆)² ways in sorting out boys and girls in case 1.

This is exactly the same in case 2, so it'd be (⁶P₆)² ways in sorting out girls and boys in case 2.

So, (⁶P₆)² + (⁶P₆)² ways = 1 036 800 ways if they alternate