A boarding house has three bedrooms and 10 students. One bedroom has 1 bed,
the second has 4 beds, and the third has 5 beds. In how many different ways
can the students be assigned rooms?


There are a number of ways to do the problem, but the answer comes out the
same, 1260, each time, even though the numbers you multiply is different: 

Choose 1 student for the room with 1 bed in 10C1 = 10 ways.
Choose 4 students for the room with 4 bed in 9C4 = 126 ways.
Choose 5 students for the room with 5 bed in 5C5 = 1 way.

10C1*9C4*5C5 = 10*126*1 ways = 1260 ways.


That's all you have to do.  However, to show you that it doesn't matter

what order you place them in the rooms, you'll always get 1260.


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Choose 1 student for the room with 1 bed in 10C1 = 10  ways.
Choose 5 students for the room with 5 beds in 9C5 = 126  ways.
Choose 4 students for the room with 4 beds in 4C4 = 1 way.

10C1*9C5*4C4 = 10*126*1 ways = 1260 ways.

 

Choose 4 students for the room with 4 beds in 10C4 = 210  ways.
Choose 1 students for the room with 1 bed in 6C1 = 6 ways.
Choose 5 students for the room with 5 beds in 5C5 = 1 way.

10C4*6C1*5C5 = 210*6*1 ways = 1260 ways.

 

Choose 4 students for the room with 4 beds in 10C4 = 210  ways.
Choose 5 students for the room with 5 beds in 6C5 = 6 ways.
Choose 1 students for the room with 1 bed in 1C1 = 1 ways.

10C1*6C4*5C5 = 10*15*1 ways = 1260 ways.

 

Choose 5 students for the room with 5 beds in 10C5 = 252 ways.
Choose 1 students for the room with 1 bed in 5C1 = 5  ways.
Choose 4 students for the room with 4 beds in 4C4 = 1 way.

10C5*5C1*4C4 = 252*5*1 = 1260 ways.


Choose 5 students for the room with 5 beds in 10C5 = 252  ways.
Choose 4 students for the room with 4 beds in 5C4 = 5  ways.
Choose 1 student for the room with 1 bed in 1C1 = 1 way.

10C5*5C4*1C1 = 252*5*1 = 1260 ways.


Edwin