Problem Solving

A sign will be painted using a certain palette consisting of 9
colors, including light blue. One of the 9 colors will be selected
at random and used to paint the border of the sign, one of the
remaining 8 colors will be selected at random and used to
paint the background of the sign, and one of the remaining 7
colors will be selected at random and used to paint the text
of the sign. What is the probability that light blue will be
chosen for the border or the background?
(A) 2/9
(B) 1/9
(C) 1/18
(D) 1/72
(E) 1/504

maihuna wrote:A sign will be painted using a certain palette consisting of 9
colors, including light blue. One of the 9 colors will be selected
at random and used to paint the border of the sign, one of the
remaining 8 colors will be selected at random and used to
paint the background of the sign, and one of the remaining 7
colors will be selected at random and used to paint the text
of the sign. What is the probability that light blue will be
chosen for the border or the background?
(A) 2/9
(B) 1/9
(C) 1/18
(D) 1/72
(E) 1/504

There are many complicated ways to solve, but we can apply simple common sense to the answer choices and solve in .001 seconds.

If the question had been "what's the probability that light blue will be chosen for the border", we'd have immediately said "1/9". Since we want the border OR the background, we've increased the chances of choosing light blue. Only (A) is greater than 1/9 - choose (A)!

that was some answer Stuart... but still how would you get 2/9?? border is light blue 1/9... but for background we will only have 8 colors....

maihuna wrote:A sign will be painted using a certain palette consisting of 9
colors, including light blue. One of the 9 colors will be selected
at random and used to paint the border of the sign, one of the
remaining 8 colors will be selected at random and used to
paint the background of the sign, and one of the remaining 7
colors will be selected at random and used to paint the text
of the sign. What is the probability that light blue will be
chosen for the border or the background?
(A) 2/9
(B) 1/9
(C) 1/18
(D) 1/72
(E) 1/504

First Approach :

By symmetry it has to be (As Stuart has pointed out ) = 1/9 (probability that border is of light blue color) + 1/9 (probability that background is of light blue color) = 2/9.

Please note that all the colors have equal probability to be either border or background or text, hence we were able to conclude the probability as 1/9 (since there are a total of 9 colors).

Another Approach :

#Total no. of outcomes = (choose border out of 9 colors)*(choose background out of remaining 8 colors)* (choose text out of remaining 7 colors) = 9*8*7

#No. of desired outcomes = [Choose light blue as border color AND choose any color from the remaining 8 colors as background AND choose any color from the remaining 7 colors as text] OR [Choose any color but light blue for border AND choose light blue as background AND choose any color from the remaining 7 colors as text]
= 1*8*7 + 8*1*7 = 2*8*7

So, probability = #No. of desired outcomes/#Total no. of outcomes = 2/9 A