Question

In a crossword puzzle, $20$ words are to be guessed of which $8$ words have each an alternative solution also. The number of possible solutions will be

• A. ${}^{20}{P}_8$
• B. ${}^{20}{C}_8$
• C. $512$
• D. $256$

Answer 1

Answer: (D)

$256$

$8$ out of the $20$ words have an alternative solution.

So $12$ words are fixed, so they won't affect the number of ways.

So now we find the number of ways in which the remaining $8$ words can be guessed.

Now every word have $2$ solutions,

so there are $2$ ways to fill each of these words, and for $8$ such words,

it can be done in $2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=2^8=256$ ways.