Problem:
An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:
the selection of four red marbles;
the selection of one white and three red marbles;
the selection of one white, one blue, and two red marbles; and
the selection of one marble of each color.
What is the smallest number of marbles satisfying the given condition?
Answer Choices:
A.
B.
C.
D.
E. more than
Solution:
The hypothesis of equally likely events can be expressed as
where , and denote the number of red, white, blue, and green marbles, respectively, and . Eliminating common terms and solving for in terms of , and , we get
The smallest for which , and are all positive integers is , with corresponding values , and . So the smallest total number of marbles is .