Question

Suppose we have 6 distinct items and we want to place 4 in one bin and 2 in the other. How many ways can this be done if order does not matter? (answer is 15)

Answer 1

We have 6 distinct items .

We want to place 4 in one bin and 2 in the other bin

To find in how many ways can this be done if order does not matter .

When order dose not matter it is combination .

We want to choose 4 balls into one bin

Number of ways of choosing 4 items from 6 distinct items into first bin = ⁶C₄

                                                                                                              = 15

Remaining items = 2

Choosing remaining 2 items in other bin = ²C₂ = 1

Thus , number of ways can this be done i.e placing 4 in one bin and 2 in the other bin

                                                                  = 15*1

                                                                  = 15

Thus there are 15 ways

Example suppose items are labeled as 1 to 6

Thus

Sr no	Bin1	Bin2
1	1,2,3,4	5,6
2	1,2,3,5	4,6
3	1,2,3,6	4,5
4	1,2,4,5	3,6
5	1,2,4,6	5,6
6	1,2,5,6	3,4
7	1,3,4,5	2,6
8	1,3,4,6	2,5
9	1,3,5,6	2,4
10	1,4,5,6	2,3
11	2,3,4,5	1,6
12	2,3,4,6	1,5
13	2,3,5,6	1,4
14	2,4,5,6	1,3
15	3,4,5,6	1,2