I believe the complete text of the question is:
Six colours (red.black.white.orange.pink.yellow) can be used to decorate. If one or more can be used, how many ways are possible that white is used?
A. 30
B. 32
C. 26
D. 400
E. 720
Well, we have to consider six cases:
(i) one color used --- there's only 1 way for that color to be white.
(ii) two colors used --- there are five other colors that can be paired with white, so in other words, 5 pairs that contain white.
(iii) three colors used --- if white is used, it will be white plus a pair chosen from five, which is calculated 5C2 = (5!)/[(2!)(3!)] = (5*4)/2 = 10. There are 10 possible triplets with white.
(iv) four colors used --- if white is used, it will be white plus a trio chosen from five, which is calculated 5C3 = (5!)/[(3!)(2!)] = (5*4)/2 = 10. There are 10 possible quartets with white.
(v) five colors used --- that means, only one of the six omitted. There's one way to omit white, and 5 ways to omit a color other than white. Thus, there are 5 quintets that contain white.
(vi) six colors used -- only one possibility, which includes white. Thus, 1 set.
Take the sum: 1 + 5 + 10 + 10 + 5 + 1 = 32
Answer = B
Does that make sense? Please let me know if you have any questions on this.
Mike
