Question

There are 10 true-false statements in a question paper. How many sequences of answers are possible in which exactly three are correct?

Answer 1

Here we will calculate the total number of possible outcomes for three correct answers keeping in mind that the order of the questions is not important.

Complete step-by-step answer:
Here, it is given to us that there are 10 true-false statements in a question paper and we need to find out the total number of possible sequences in which exactly three answers are correct.
The first correct answer could be anyone 1 question out of the 10 questions.
The second correct answer could be anyone 1 question out of the 9 questions remaining.
Similarly, the third correct answer could be anyone 1 question out of the 8 questions remaining.
Since the order of the answers is not important so they can occur in 3! ways. That is the first answer could be anyone out of the 3 correct answers, the second answer could be any one out of the 2 correct answers left, and the third answer is the remaining correct answer.
The total number of possible sequences are
$= \dfrac{{Total\,number\,of\,possible\,sequences\,of\,all\,answers}}{{Total\,number\,of\,possible\,sequences\,of\,correct\,answers}}$
$= \dfrac{{10 \times 9 \times 8}}{{3 \times 2 \times 1}} = 120\,$ sequences are possible with exactly three correct answers.

Note: This question can also be solved using combination as order of the 3 correct answers is not important. Combination is defined as $^n{C_r}$, where n is the total number of outcomes and r is the number of favourable or desired outcomes.
$^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}$
Here, n is 10 and r is 3. So
$^{10}{C_3} = \dfrac{{10!}}{{\left( {10 - 3} \right)!3!}}$
$^{10}{C_3} = \dfrac{{10!}}{{7!3!}}$
$^{10}{C_3} = \dfrac{{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{\left( {7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \right) \times 3 \times 2 \times 1}} = 120$ sequences are possible with exactly three correct answers.
If in the question some condition for order of the correct answers would have been also mentioned then instead of combination, permutation would have been used.