Question

An examination consists of 10 multiple choice questions, where each question has 4 options, only one of which is correct. In every question, a candidate earns 3 marks for choosing the correct opion, and -1 for choosing a wrong option. Assume that a candidate answers all questions by choosing exactly one option for each. Then find the number of distinct combinations of anwers which can earn the candidate a score from the set {15, 16,17,18, 19, 20}.

Answer 1

Correct Answer - 120
Suppose he answers x questions correctly and 10-x question wrongly.
`therefore` His score is `3x+(10-x)(-1)=4x-10`
Now score of the candidate is from the set {15,16,17,18,19,20}.
Only 4x-10=18 is possible .
`therefore x=7`
So, his 7 answer are correct and 3 answers are wrong.
Number of ways of selection of 7 questions from `10= .^(10)C_(7)= .^(10)C_(3)`
`=(10xx9xx8)/(6)=120`