The yearbook committee has to

[ChristopherR703]

I'm having trouble understanding the logic behind the more complicated combinatorics problems. Consider this question from page 62 in the 6th edition guide 5 Number Properties book:

The yearbook committee has to pick a color scheme for this year's yearbook. There are 7 colors to choose from (red, orange, yellow, green, blue, indigo, and violet). How many different color schemes are possible if the committee can select at most 2 colors?

My thoughts: so we can have 1 or 2 colors since the problem states 'at most.' If we choose 1 color only that would be 7 schemes. But, if we chose 2 colors, we would have 7 colors to choose for the first spot and 6 for the second. 7x6 is 42 schemes with 2 colors. How is that wrong? The explanation in the guide shows 7!/2!5!. Why in the world do we care about the colors not chosen!!

[RonPurewal]

if there is a distinct order to the two colors—i.e., if there are distinct 'first' and 'second' colors (or 'background'/'foreground', or any other such distinction)—then your answer is correct.

on the other hand, if it's just 'let's pick two colors', then your 7x6 is counting every possibility twice. in that case the number of pairs of colors is not 7x6 but (7x6)/2 = 21.

[RonPurewal]

by the way, if anything like this shows up on the official exam, the problem will be very painstakingly explicit about whether the colors have two distinct roles (= whether 'order matters').

so if this problem was not explicit enough about that, then you can rest easy.