An jar is filled with red, white, and blue tokens that are e

ziyuenlau wrote:A jar is filled with red, white, and blue tokens that are equivalent except for their color. The chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token. If the number of tokens of every color is a multiple of 3, what is the smallest possible total number of tokens in the jar?

(A) 9
(B) 12
(C) 15
(D) 18
(E) 21

We can PLUG IN THE ANSWERS, which represent the total number of tokens.
When the correct answer choice is plugged in, P(red then white) = P(blue).
Since the question stem asks for the SMALLEST POSSIBLE TOTAL, start with the smallest answer choice.

A: 9 tokens, implying 3 red, 3 white, 3 blue
P(red then white) = 3/9 * 3/9 = 1/9.
P(blue) = 3/9 = 1/3.
Since P(red then white) < P(blue), eliminate A.

To increase the value of P(red then white), R+W must constitute a greater portion of the total.
Thus, as we test the remaining answer choices, we should keep increasing the value of R+W.

B: 12 tokens, allowing for 6 red, 3 white, 3 blue
P(red then white) = 6/12 * 3/12 = 1/8.
P(blue) = 3/12 = 1/4.
Since P(red then white) < P(blue), eliminate B.

C: 15 tokens
Case 1: 9 red, 3 white and 3 blue, in which case P(red then white) = 9/15 * 3/15 = 3/25.
Case 2: 6 red, 6 white and 3 blue, in which case P(red then white) = 6/15 * 6/15 = 4/25.
P(blue) = 3/15 = 1/5.
In each case, P(red then white) < P(blue).
Eliminate C.

Notice that Case 2 -- in which R=W -- yields a greater result than Case 1.
Thus, in D and E, we should test cases in which the values for R and W are as close as possible.

D: 18 tokens, allowing for 9 red, 6 white, 3 blue
P(red then white) = 9/18 * 6/18 = 1/6.
P(blue) = 3/18 = 1/6.
Success!

The correct answer is D.