Your question is very interesting. It combines a lot of simple (but very easy to miss on test day) combiniatorics issues.
First we have 2 Sizes. Lets for simplicity's sake call them LARGE and SMALL. Then there's 4 colors in each kind.
Now the question wants us to generate all the ways in which we can package (Same Color AND Same Size) OR (Same Size and 3 different colors). The AND and the OR are important here. The OR tells us to add the 2 different numbers arrived at.
Lets do the Same Color AND Same Size first :
Since there must be 3 notepads of the same color, it could be either:
BBB
GGG
YYY
PPP
( B = Blue, G= Green, Y= Yellow and P= Pink). Thats 4 ways.
But, we have to account for the 2 sizes also. And since it is an "AND" condition, multiply 4 with 2 = 8 ways.
Now, let's do the Same Size AND Different Colors:
The number of ways to pick 3 different colors from 4 given colors (when order is not important) is 4 C 3. This combination gives us 4 ways.
But, to account for the 2 sizes, we multiply this by 2. Again, the "AND" in the condition is key.
Now , we have an "OR", which in combinatorics is an addition.
So, 8 + 8 = 16 ways in which to fulfill the 2 given conditions of filling these boxes.
Hope this helps!
For love, not money.
