JEE Main · 2023 · 08 Apr 2023 · Shift 1 · Permutations and Combinations

First arrange the 7 boys in a circle, which can be done in $(7-1)!=6!$ ways. This creates 7 gaps between the boys. We need to select 5 of these 7 gaps to place the girls, which can be done in $\left(\genfrac{}{}{0ex}{}{7}{5}\right)$ ways. The 5 girls can be arranged in these 5 gaps in $5!$ ways. Total arrangements = $6!\times \left(\genfrac{}{}{0ex}{}{7}{5}\right)\times 5!=6!\times 21\times 5!=720\times 21\times 120=126\times 120^2=126(5!{)}^2$.