How many possible answers are there?

Question

i have a test that allows you to pick multiple answers per question. There are 4 options to choose from, and u don't know if the right answer is all of them, 3, 2, or just one of them.  So how many combinations are possible???

 

granted you can assume if it was just 1 answer it would have been the type on the test where you only pick 1…. Irregardless, I'm just curious how many options it could be?   I was thinking 4 to the 4th to the 4th maybe???

Answer 1

Think about it.  There is one way (1234) to choose four out of four, four ways (123, 124, 134, 234) to choose three out of four, six ways (12, 13, 14, 23, 24, 34) to choose two out of four, and four ways (1, 2, 3, 4) to choose one out of four.  Fifteen possible combinations in all.

You can also use factorials to answer this question without listing out the possible combinations.  If n is the total number and x is the number chosen, then...

 

nCx = n! / (x! (n-x)!) 

 

4C4 = 4! / (4! 0!) = 1.  Remember that 0! = 1.

 

4C3 = 4! / (3! 1!) = 4.

 

4C2 = 4! / (2! 2!) = 6.

 

4C1 = 4! / (1! 3!) = 4.

 

1 + 4 + 6 + 4 = 15.

 

The remaining possibility - choosing zero out of four - can be disregarded if you must choose at least one of the four answers.