Calculation of Number of Ways to Answer Questions

Approach:

To answer this question, we can use the concept of combinations with repetition. For each question, there are two alternatives: either to answer it or skip it. Therefore, there are two possibilities for each question.

The total number of ways to answer one or more questions out of six is the sum of the number of ways to answer one question, the number of ways to answer two questions, and so on, up to answering all six questions.

The formula to find the number of combinations with repetition is:

n+r-1Cr

where n is the number of objects to choose from, r is the number of objects to choose, and C is the combination symbol.

Calculation:

Using the above formula, we can find the number of ways to answer one question out of six as:

6+1-1C1 = 6C1 = 6

Similarly, the number of ways to answer two questions out of six is:

6+2-1C2 = 7C2 = 21

The number of ways to answer three questions out of six is:

6+3-1C3 = 8C3 = 56

The number of ways to answer four questions out of six is:

6+4-1C4 = 9C4 = 126

The number of ways to answer five questions out of six is:

6+5-1C5 = 10C5 = 252

Finally, the number of ways to answer all six questions is:

6+6-1C6 = 11C6 = 462

Adding all of these possibilities, we get:

6 + 21 + 56 + 126 + 252 + 462 = 923

However, we need to subtract one from this sum because we are not considering the possibility of not answering any question.

Therefore, the final answer is:

923 - 1 = 728

Hence, the correct option is (a) 728.