Mathematics - Permutation & Combination Question with Solution | TestHub

Consider exactly three persons between the brothers as reference now second brother can be placed in two ways (left or right), rest 18 in ${}^{18!}$ Ways.

Similarly we have $2\times 18!$ arrangements for exactly 4 persons or 5 persons or 6 persons or 7 persons or 8 persons in between the brothers. For exactly 9 persons in between them we have only 18! Ways. These will also include 10 or 11 or 12 or 13 or 14,15 persons in between the brothers. So total number of arrangements $=6\times 2\times 18!+18!=13\times 18!$ Alternate If one of the brothers is made reference point then remaining 18 persons (excluding the second brother) can be seated in 18! Ways. For the second brother we have only $19-6=13$ places to sit.

⇒ Total number of ways $=13\times 18!$