Chapter 12.6, Problem 65PPE

a.

To determine

To calculate: The number of ways student could choose 4 of the first 6 questions.

Answer to Problem 65PPE

There are $15$ ways to choose 4 questions.

Explanation of Solution

Given information:

A test has 10 questions. Student must answer a total of 7 questions , including exactly 4 of the first 6 questions.

Formula used:

A combination is a selection of objects without regard to order. The expression $C{}_r$ represents the number of combination of $n$ objects chosen $r$ at a time.

  $C{}_r=\frac{n!}{r!\left(n-r\right)!}$

For any positive integer $n$, the expression $n$ factorial is written as $n!$ and is the product of integers from n down to 1.

Calculation:

According to the question,

Test has total 10 questions.

Student has to answer total 7 questions , including exactly 4 of the first 6 questions.

Now,

Questions to be chosen out of first six questions are 4.

  $\begin{array}{l}C{}_4=\frac{6!}{4!\left(6-4\right)!}\\ =\frac{6!}{4!2!}\\ =\frac{6\times 5\times 4\times 3\times 2\times 1}{4\times 3\times 2\times 1\times 2\times 1}\\ =\frac{6\times 5}{2\times 1}\\ =3\times 5\\ =15\end{array}$

Hence, there are $15$ possible ways to choose 4 questions out of first six questions.

b.

To determine

To calculate: The number of questions left after answering 4 out of first 6 questions and the number of questions student can still answer.

Answer to Problem 65PPE

There are $6$ questions to be answered and $3$ questions are still to be answered.

Explanation of Solution

Given information:

A test has 10 questions. Student must answer a total of 7 questions , including exactly 4 of the first 6 questions.

Formula used:

Subtraction is used.

Calculation:

According to the question,

Test has total 10 questions.

Student has to answer total 7 questions , including exactly 4 of the first 6 questions.

Now,

Number of questions done out of first six questions are 4 and total questions are 10.

  $\begin{array}{l}=10-4\\ =6\end{array}$

Also, the number of questions supposed to be done are 7 and 4 have already answered.

The number of questions left are

  $\begin{array}{l}=7-4\\ =3\end{array}$

Hence, there are $6$ questions to be answered and $3$ questions are still to be answered.

c.

To determine

To calculate: The number of ways to choose the questions.

Answer to Problem 65PPE

There are $20$ ways to finish the test.

Explanation of Solution

Given information:

A test has 10 questions. Student must answer a total of 7 questions , including exactly 4 of the first 6 questions.

Formula used:

A combination is a selection of objects without regard to order. The expression $C{}_r$ represents the number of combination of $n$ objects chosen $r$ at a time.

  $C{}_r=\frac{n!}{r!\left(n-r\right)!}$

For any positive integer $n$, the expression $n$ factorial is written as $n!$ and is the product of integers from n down to 1.

Calculation:

According to the question,

Test has total 10 questions.

Student has to answer total 7 questions , including exactly 4 of the first 6 questions.

Now,

Number of questions done out of first six questions are 4 and total questions are 10.

  $\begin{array}{l}=10-4\\ =6\end{array}$

Questions to be chosen from rest of the paper are 3.

Therefore,

  $\begin{array}{l}C{}_3=\frac{6!}{3!\left(6-3\right)!}\\ =\frac{6!}{3!3!}\\ =\frac{6\times 5\times 4\times 3\times 2\times 1}{3\times 2\times 1\times 3\times 2\times 1}\\ =\frac{6\times 5\times 4}{3\times 2\times 1}\\ =5\times 4\\ =20\end{array}$

Hence, there are $20$ possible ways to finish the test.

d.

To determine

To calculate: The number of ways student can complete the test.

Answer to Problem 65PPE

There are $300$ ways to complete the test.

Explanation of Solution

Given information:

A test has 10 questions. Student must answer a total of 7 questions , including exactly 4 of the first 6 questions.

Formula used:

Multiplication is used.

Calculation:

According to the question,

Test has total 10 questions.

Student has to answer total 7 questions , including exactly 4 of the first 6 questions.

Now,

There are $15$ possible ways to choose 4 questions out of first six questions.

There are $20$ possible ways to finish the test.

Therefore,

  $\begin{array}{l}=15\times 20\\ =300\end{array}$

Hence, there are $300$ ways to complete the test.