Question

A test containstrue/false questions. How many different ways can a student answer the questions on the test, if answers may be left blank?

Short answer

The number of ways a student can answer the questions on the test, if answers may be left blank is \[{\rm{515,377,520,732,011,331,036,461,129,765,621,272,702,107,522,001}}\] ways.

Step-by-step solution

Concept Introduction

Definition of permutation (order is important) is –

No repetition allowed:\[{\rm{P(n,r) = }}\frac{{{\rm{n!}}}}{{{\rm{(n - r)!}}}}\]

Repetition allowed:\[{{\rm{n}}^{\rm{r}}}\]

Definition of combination (order is important) is –

No repetition allowed:\[{\rm{C(n,r) = }}\frac{{{\rm{n!}}}}{{{\rm{r!(n - r)!}}}}\]

Repetition allowed:\[{\rm{C(n + r - 1,r) = }}\frac{{{\rm{(n + r - 1)!}}}}{{{\rm{r!(n - 1)!}}}}\]

With\[{\rm{n! = n}} \cdot {\rm{(n - 1)}} \cdot ... \cdot {\rm{2}} \cdot {\rm{1}}\].

Finding the number of ways

Order is important (since a different order of the answers to the questions will lead to a different test result), thus it is needed to use permutation.

It is needed to select\({\rm{100}}\)answers from the possible\({\rm{3}}\)answers (true, false and blank).

Here, it can be seen\({\rm{n = 3, r = 100}}\).

Since repetition is allowed (as multiple answers can be true), so substitute the value and calculate –

\(\begin{array}{c}{{\rm{n}}^{\rm{r}}}{\rm{ = }}{{\rm{3}}^{100}}\\{\rm{ = 515,377,520,732,011,331,036,461,129,765,621,272,702,107,522,001}}\end{array}\)

Therefore, the result is obtained as \({\rm{515,377,520,732,011,331,036,461,129,765,621,272,702,107,522,001}}\).