Permutations & Combinations
Queue Arrangements with Block and Exclusion Conditions
nta_pyq_2025_apr
Grade 11
Question:
3 girls and 4 boys (including $B_1$ and $B_2$) stand in a queue. The number of arrangements where all girls stand together and all boys stand together, but $B_1$ and $B_2$ are not adjacent, is
Step-by-Step Solution
Key Concept: Count total arrangements with boys-block and girls-block (= $2!\cdot3!\cdot4!$), then subtract those where $B_1$ and $B_2$ are also adjacent within the boys-block.
Total (all girls together, all boys together): $2!\cdot3!\cdot4!=2\cdot6\cdot24=288$.
Subtract ($B_1$ and $B_2$ also adjacent within boys-block): treat $\{B_1,B_2\}$ as one unit, boys-block has $3!$ arrangements of the unit and $B_3,B_4$, and $B_1B_2$ can be internally ordered in $2!$ ways:
$2!\cdot3!\cdot3!\cdot2!=2\cdot6\cdot6\cdot2=144$.
Answer: $288-144=144$.
Correct Answer: 2