Date: Thu, 19 Sep 1996 20:37:17 -0600 (CST) Sender: David Subject: statistical (?) query Name: David Who is asking: Student Level: AllQuestion:
Suppose there is a 12 question test. Each question can be answered only by true or false. Assuming that all questions are answered (i.e. none are left blank), how many possible combinations of answers are there? And what formula was used to arrive at the answer?
Everyone I ask has a different answer (12 x 12, 12 to the power of 12, etc). Your answer is greatly appreciated!
The reasoning is that you have 2 ways to perform the first task -- answer question 1 T or F; for each of these two ways you have 2 ways to perform the 2nd task -- answer question 2 T or F for now a total of 4 combinations; for each of these 4 ways you have 2 ways to perform the 3rd task -- answer question 3 T or F for now a total of 4 times 2 = 8 combinations; for each of these 8 ways you have 2 ways to perform the 4th task -- answer question 4 T or F for now a total of 8 times 2 = 16 combinations; and so on until we get to 2^12 ways by the time we've finished #12.
It may be a bit more transparent to see the multiplication principle if you plan on getting dressed using one of 4 hats, one of 5 shirts, one of 2 pants, and one of 7 pairs of sandals -- how many different 'outfits' can you make? For each of 4 hats you combine this with one of 5 shirts for 20 possibilities so far. When you add pants you now have 40 possibilities and each of these can be matched up with one of 7 pairs of sandals for a total of 4 x 5 x2 x 7 = 280 outfits.
In general if you are to perform k tasks and they can be done in n1, n2, ... nk ways respectively the total number of ways of completing these tasks is n1 x n2 x ... x nk.
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