Name: Salman

Question:
a travel agency has limited number of six different free pamphlets about the beetle islands. the agent tells you to take any that you want, but not more than two of any kind. Assuming that you take at least one pamphlet, how many choices could you make

For the moment disregard the requirement that you must take at least one pamphlet. In front of you are 6 stacks of pamphlets. From the first stack you can take either 0, 1 or 2 pamphlets. Thus for the first stack alone you have 3 choices.

Once you have made that choice go to the second stack. Regardless of your choice for the first stack, you have 3 choices for the second stack, you can take 0, 1 or 2 pamphlets. Thus for the first two stacks you have 3 x 3 = 9 choices.

Now go to the third stack. Again you have 3 choices from this stack, so for the first three stacks you have 3 x 3 x 3 = 3³ = 27 choices.

Continuing in this fashion, for 6 stacks you have 3⁶ choices in total.

Now consider the the requirement that you must take at least one pamphlet. Of the 3⁶ possible choices counted above, only one does not satisfy this condition. That is where you chose zero pamphlets from each of the six stacks. Hence the number of choices if you must take at least one pamphlet is 3⁶ - 1.