Question
Question: Out of a crew of 43 persons on a ship, only 3 can do the steering and one is needed. Of the remainin...
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?
Solution
First select 1 person from the three persons who can steer the ship using the formula of combinations 3C1. Now, consider that row 1 has already 8 rowers and row 2 has 3 rowers. Select the remaining 12 rowers from 29 rowers left needed for row 1 by using the relation 29C12. The remaining 17 will automatically get selected for row 2. Finally, arrange the 20 rowers among themselves present in each row by using the relation 20!×20!. Multiply all the expressions to get the answer.
Complete step by step answer:
First of all we need to select 1 person who will steer the ship out of three persons who can do the steering. So we have,
⇒ Number of ways to select 1 person from 3 persons for steering = 3C1
Now, it is said that there should be 20 rowers on the sides of the boat. Also, 8 rowers have their own wish to go to one side and 3 rowers have their wish to go on the other. Let us assume that 8 rowers are in row 1 and 3 are in row 2. So, 29 rowers are left in which row 1 must have 12 more rowers and the remaining 17 must be in row 2.
⇒ Number of ways to select 12 persons from 29 persons for row 1 = 29C12
Therefore the remaining 17 will automatically get selected for row 2.
So we have selected the rowers for both sides, now we need to arrange them. So we get,
⇒ Number of ways in which 20 rowers can be arranged among themselves in row 1 = 20!
⇒ Number of ways in which 20 rowers can be arranged among themselves in row 2 = 20!
Therefore, the total number of arrangements that can be made = 3C1×29C12×20!×20!
Note: Note that we haven’t applied the combination formula for the selection of 17 people that is needed in row 2 because we have already selected 12 rowers out of 29 for row 1 and therefore the remaining rowers are 17 in which we have to select 17 so there will be only one way of selection given as 17C17. The mathematical reason for such a condition is due to the formula nCr=nCn−r.