Question

A professor writes \(40\) discrete mathematics true/false questions. Of the statements in these questions, \(17\) are true. If the questions can be positioned in any order, how many different answer keys are possible?

Short answer

The total number of possible ways is \(88,732,378,800\).

Step-by-step solution

Given data

Number of true/false questions \( = 40\) and number of true questions \( = 17\).

Concept of Combination

A combination is a selection of items from a set that has distinct members.

Formula:

\(_n{C_r} = \frac{{n!}}{{r!(n - r)!}}\)

Calculation for the number of answer keys

Find the number of different possible answer keys.

Number of possible ways \( = C(40,17)\)

\(\begin{array}{l}C(40,17) = \frac{{40!}}{{17!(40 - 17)!}}\\C(40,17) = \frac{{40!}}{{17!23!}}\\C(40,17) = 88,732,378,800\end{array}\)

Hence, the total number of possible ways is \(88,732,378,800\).