Question

(a) On a 10-question test, how many ways are there to answer exactly eight

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.