Permutation and combination

Question

In a cross word puzzle 20 words are to be guessed of which 8 words have each an alternative solution.the number of possible solution is ?

 

Answer 1

In a cross word puzzle 20 words are to be guessed of which 8 words have each an alternative solution. the number of possible solution is ?

 

A "solution" to this crossword puzzle has the 12 (that is, 20-8) words with their one and only answer.

 

The other 8 words may have either of their respective 2 alternatives.

 

Thus, the number of possible solutions is 2*2*2*2*2*2*2*2  (consider each alternative for each of the 8 words). That makes 256 possible solutions.

 

If you were to list the solutions, it would look like:

  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15   16  17  18  19  20

  x  x  x  x  x  x  x  x  x   x     x    x    A   A    A     A    A   A    A    A

  x  x  x  x  x  x  x  x  x   x     x    x    A   A    A     A    A   A    A    B

  x  x  x  x  x  x  x  x  x   x     x    x    A   A    A     A    A   A    B    A

  x  x  x  x  x  x  x  x  x   x     x    x    A   A    A     A    A   A    B    B

  x  x  x  x  x  x  x  x  x   x     x    x    A   A    A     A    A   B    A    A

  x  x  x  x  x  x  x  x  x   x     x    x    A   A    A     A    A   B    A    B

 . . .

  x  x  x  x  x  x  x  x  x   x     x    x    A   B    B     B    B   B    B    B

 

This is very much like a "counter" (like an odometer in a car or a gas pump counter) that goes from 0-9 for each digit.  Note that from 0000 to 9999, there are 10*10*10*10=1000 possibilities.

Answer 2

Let me try to understand this problem... crossword puzzle normally each word has its own answer. However, in this case, 8 of those answers can be answered with another possibility. So, for the first 8 answers, those can be answered with either an A choice or a B choice. That part would be 8*2! or 16. The second section, the remaining 12 answers, each have one possible answer. This would give 12 answers. So, 16 + 12 = 28.