Question

How many ways are there to place 7 distinct objects into 5 non-empty identical bins?

• A. 140
• B. 156
• C. 152
• D. 148

Answer 1

The correct option is A 140

The objects will either be in a distribution of 1,1,1,1,3 or 1,1,1,2,2.
Note that it does not matter what order the objects are in, only which objects are grouped together. For the first case, there are ${}^7{C}_3$ ways to choose the grouping of 3 objects. Then, in the second case, there are ${}^7{C}_4$ ways to select 4 objects out of the 7, and then there are $\frac{{}^4{C}_2}{2}$ = 3 ways to divide these 4 objects into 2 groups of 2. The total number of ways to arrange the objects is ${}^7{C}_3+3{\ast }^7{C}_4=140$