JEE Main 2026 — Permutation Combination Question with Solution

Total number of boys $=4$

Total number of girls $=3$

Total number of children $=7$

Total number of ways to arrange $7$ children in a queue $=7!=5040$

To find the number of ways where all $3$ girls are together, consider the $3$ girls as a single unit.

Total number of units to arrange $=4$ boys $+1$ unit of girls $=5$ units.

Number of ways to arrange these $5$ units $=5!=120$

Number of ways to arrange the $3$ girls among themselves $=3!=6$

Total number of ways where all girls are together $=5!\times 3!=120\times 6=720$

Number of ways such that all the girls are not together $=\text{Total ways}-\text{Ways where all girls are together}$

$=5040-720=4320$

Answer: $4320$