Glue them together, or keep them apart.

A row of seats: glue chosen people into a block (arrange block × inside), or seat one group first and drop the others into the marked gaps including both ends.

With 4 girls and 3 boys, count arrangements with no two boys together and get 1440 (4! × 5P3).

Forgetting arrangements inside a block; ignoring the two end gaps.

• A leading zero makes a number shorter, so it must be removed. Three-digit numbers from the digits 0, 1, 2, 3, 4 without repetition: all arrangements 5P3 = 60, minus those starting with 0, 4P2 = 12, leaves 48. Directly: 4 × 4 × 3 = 48.
• Fixed places first: four-digit numbers greater than 3000 from the digits 1 to 5 without repetition must start with 3, 4 or 5, giving 3 × 4P3 = 3 × 24 = 72.
• Even three-digit numbers from 1, 2, 3, 4, 5 without repetition: the units digit is 2 or 4 (2 ways), then 4 and 3 ways for the other places, so 2 × 4 × 3 = 24.
• Together: tie the objects that must stay together into one block, arrange the block with the others, then arrange inside the block. RELAY with its vowels E and A together: 4 units in 4! = 24 ways, times 2! inside the block, gives 48.
• Not together: subtract from the total. RELAY has 5! = 120 arrangements, so the vowels are apart in 120 − 48 = 72 of them.
• Apart (gap method): seat 4 girls first in 4! = 24 ways; they leave 5 gaps, including the two ends. Seat the 3 boys in 3 of these gaps in 5P3 = 60 ways. No two boys are then together, and the total is 24 × 60 = 1440.

Teaching this idea? Put the simulation on your own page or LMS with this code.