Question

Out of a crew of 43 persons on a ship, only 3 can do the steering and one is needed. Of the remaining 40, 8 persons wish to go only on one side and 3 persons wish to go on the other side. In how many ways can they be arranged so that there are 20 rowers on each side of the ship?

Answer 1

${}^3{C}_1$ = 3 for steering.
Out of 40 rowers 8A 3B (by choice)
Remaining 29 12A 17B (by selection)
This selection can be done in ${}^{29}{C}_{12}$. ${}^{17}{C}_{17}$
Thus total selections are ${}^3{C}_1$. ${}^{29}{C}_{12}$. ${}^{17}{C}_{17}$
Now 20 rowers on side A can be arranged amongst themselves in 20 ! ways and similarly on side B. Hence total number is
${}^3{C}_1$. ${}^{29}{C}_{12}$. ${}^{17}{C}_{17}$.20! 20!