I have four identical oranges. How many ways are there for me to divide these oranges into at most three groups? (By definition, a group must have at least one orange.)
+128707 "At most" connotes 1, 2 or 3 groups
For one group = one way....all of them together
For two groups.....2 + 2 or 3 + 1 = two ways ...{the oranges are indistinguishable, so 1 + 3 is the same grouping as 3 + 1 }
For 3 groups = 1 + 1 + 2 = one way {again, they are indistinguishable}
So ..... four ways.....
A Senate committee has 5 Democrats, 5 Republicans, and 1 Independent. In how many ways can they sit around a circular table if all the members of each party all sit next to each other? (Two seatings are considered equivalent if one is a rotation of the other.)
+128707 "At most" connotes 1, 2 or 3 groups
For one group = one way....all of them together
For two groups.....2 + 2 or 3 + 1 = two ways ...{the oranges are indistinguishable, so 1 + 3 is the same grouping as 3 + 1 }
For 3 groups = 1 + 1 + 2 = one way {again, they are indistinguishable}
So ..... four ways.....
