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

Hint:In this question, we first need to find how many words out of those given 20 words should be guessed. Now, after removing 8 words that have alternative options the other 12 words will be fixed. Now, each word in these 8 words has 2 ways. then, on multiplying all these ways we get the total number of ways.

Complete step-by-step answer:
Now, from the given conditions in the question 8 words out of 20 given words have alternate options
Here, 12 words out of the given 20 words are fixed and they don't really effect the total number of ways.
Now, we need to find the number of ways in which the remaining 8 words can be guessed.
Now, each of these 8 words have 2 solutions to be guessed
Now, the numbers of ways in which each word can be guessed is given by
\[\Rightarrow 2\text{ ways}\]
So, as there are 8 such words now we can find the total number of ways as
\[\Rightarrow \text{2}\times \text{2}\times \text{2}\times \text{2}\times \text{2}\times \text{2}\times \text{2}\times \text{2}\]
Now, this can be further written as
\[\Rightarrow {{\text{2}}^{8}}\]
Now, on further simplification we get,
\[\Rightarrow 256\]
Hence, the correct option is (d).

Note: It is important to note that the 8 words have alternate options and the other 12 words are fixed. So, those 12 words don't need to be guessed as they don't have alternative options and the result won't be changed because of those 12 words.It is also to be noted that each word out of those 8 words has 2 ways as they have an alternative option which means one more way extra. Here, neglecting any of the words or any of the ways changes the result completely.