In how many ways can 5 boys and 3 girls stand in a row so that no two girls are next to each other?
14,400. Arrange the boys first: 5! = 120. They create 6 gaps (including the two ends). Choose and fill 3 of those gaps with the girls in order: 6 x 5 x 4 = 120. Total 120 x 120 = 14,400.
Explain it simply
To keep certain people apart, seat the others first and then place the "keep apart" people in the gaps between them, like spacing out the strongest batters in a batting order. With 5 boys in a row there are 6 gaps: one before each boy and one at the end. Putting at most one girl per gap guarantees no two girls touch.
Example
Question: 5 boys, 3 girls in a row, no two girls adjacent.
- Arrange boys: 5! = 120.
- Gaps: B B B B B = 6 gaps.
- Place 3 distinct girls in 3 of the 6 gaps: P(6, 3) = 6 x 5 x 4 = 120.
- Total = 120 x 120 = 14,400.
Contrast: all 3 girls together would be 6! x 3! = 720 x 6 = 4,320. "No two together" is stricter than "not all together", which would be 8! - 4,320 = 40,320 - 4,320 = 36,000.
const f = n => (n <= 1 ? 1 : n * f(n - 1));
console.log(f(5) * 6 * 5 * 4); // Output: 14400
console.log(f(8) - f(6) * f(3)); // Output: 36000What to say in the interview
"I seat the 5 boys first, 120 ways, which leaves 6 gaps. Putting the 3 girls into different gaps is 6 times 5 times 4, which is 120. So 14,400 arrangements."
Common mistakes
- Using 4 gaps (only between boys) and forgetting the two ends.
- Confusing 'no two together' with 'not all three together'.