Question

If there are 7 boys and 5 boys and 5 girls in a class, then in how many ways they can be seated in a row such that 

(i) No two girl sit together? 

(ii) All the girls never sit together?

Answer 1

(i) Total girls (G) = 5 

Total boys (B) = 7 

G B G B G B G B G B G B G B G 

Hence possible places of (G) girls = 8 

∴ The total number of ways that 5 girls occupy 8 

Place = ⁸P₅ = \(\frac{8!}{3!}\) = 6720 

∴ Total number of ways no two girls sit together = ⁸P₅ × 7! 

(ii) There are total 12 girls and boys, so we can arrange them in! ways. 

But when all the girls sit together then, we treat then as single place. So we have 8 place including (7 boys + 1 place for all 5 girls), which can be arranged in 8! Ways. 

8! (12 ∙ 11 ∙ 10 ∙ 9 − 5!) 

8! (11880 − 120) 

8! × 11760 

= 11760 × 8! 

Hence, total required ways 

= 12! − 8! 5!