Question

(a) On a 10-question test, how many ways are there to answer exactly eight
questions correctly?
(b) Repeat part (a) with the requirement that the first or second question, but
not both, are answered correctly.
(c) Repeat part (a) in the case that three of the first five questions are answered
correctly.

Answer 1

Answer:

(a) There are 45 ways to answer exactly 8 questions correctly.

(b) There are 16 ways to answer exactly 8 questions correctly such that either 1st or 2nd is correct.

(c) There are 10 ways to answer exactly 8 questions correctly such that the 3 of the first 5 questions are correct.

Step-by-step explanation:

(a)

Combination is the procedure to select k items from n distinct objects.

${n\choose k}=\frac{n!}{k!(n-k)!}$

The total number of questions in the test is, 10.

The number of questions answered correctly is, 8.

Compute the combination of 8 questions from 10 as follows:

${10\choose 8}=\frac{10!}{8!(10-8)!}=\frac{10!}{8!\times 2!}=\frac{9\times10}{2\times1}=45$

Thus, there are 45 ways to answer exactly 8 questions correctly.

(b)

Now a condition is applied that of the first two questions only one is correct not both.

The sample space of selecting 8 correctly answered questions from 10 is:

S = {1st is correct and remaining 7 can be selected from the rest 8,

      2nd is correct and remaining 7 can be selected from the rest 8}

• Number of ways to select 7 correct questions from the remaining 8 given that the 1st question is correct and 2nd is wrong is:

        ${8\choose 7}=\frac{8!}{7!(8-7)!}=8$

• Number of ways to select 7 correct questions from the remaining 8 given that the 1st question is wrong and 2nd is correct is:

        ${8\choose 7}=\frac{8!}{7!(8-7)!}=8$

The total number of ways to select 8 correctly answered questions such that either 1st or 2nd is correct is = 8 + 8 = 16.

Thus, there are 16 ways to answer exactly 8 questions correctly such that either 1st or 2nd is correct.

(c)

The condition now applied is that the 3 of the first 5 questions are correct.

Number of ways to select 3 correct answers from the first 5 questions is:

${5\choose 3}=\frac{5!}{3!(5-3)!}=\frac{5!}{3!\times2!}=10$

Number of ways to select 5 correct answers from the last 5 questions is:

${5\choose 5}=\frac{5!}{5!(5-5)!}=\frac{5!}{5!\times0!}=1$

The total number of ways to select 8 correctly answered questions such that the 3 of the first 5 questions are correct is = 10 × 1 = 10.

Thus, there are 10 ways to answer exactly 8 questions correctly such that the 3 of the first 5 questions are correct.