Question

1. A special at a local restaurant allows customers to choose one entrée and one dessert for $5.99. There are 8 entrées and 5 desserts to choose from. Use the Fundamental Counting Principle to calculate the total number of specials available.
A.8
B.13
C.26
D.40

2. There are 30 red soda cans and 20 green soda cans in an ice chest. What are the odds of reaching in and grabbing a green can?
A.3/2
B.2/3
C.3/5
D.2/5

3. A coin is flipped, then a 6-sided die is rolled. What is the probability of getting heads and an even number?
A.1/2
B.1/4
C.1/6
D.1/12

4. Using the letters in the word INNOVATIVE, find the number of permutations that can be formed using 4 letters at a time. Show your work or explain how you got your answer.

5. The gym has 11 different types of machines in the weight room. Geoff has time to use only 3 of them this afternoon. How many different combinations of machines can Geoff choose from to use? Show your work or explain how you got your answer.

6. List the sample space for flipping a two-sided coin two times.

Answer 1

Answer:

1-D)40

2-D $\frac{2}{5}$

3-B) $\frac{1}{4}$

4 There are 120 permutations.

5 There are 165 combinations.

6 Sample space is {HH,HT,TH,TT}

Step-by-step explanation:

Number 1

Step 1

The fundamental counting theorem states that for  process that can be carried out in k steps where the fist step can be done in  $n_1$ ways, step 2 can be done in $n_2$  number of ways and the  $k^{th}$ can be done in $n_k$ number of ways, the number of ways to complete this task is $n_1\times n_1\times....\times n_k$ ways.

Step 2

We now realize that this process can be carried out in 2 steps , there are 8 ways to complete the first step and 5 ways to complete the second step. The number of ways to carry out this calculation is shown below,

$8\times 5=40.$ The correct answer is D.

Number 2

Step 1

in this step we calculate the number total number of cans in the ice chest.  Since there are 30 red cans and 20 green cans, there is a total of 50 cans.

Step 2

In this step we find the probability of grabbing a green can by dividing the total number of green cans by the total number of cans. The calculation for the probability is shown below,

$P(G)=\frac{20}{50} =\frac{2}{5}.$

The correct answer is D.

Number 3

Step 1

The first step is to realize that  there is a $\frac{1}{2}$ chance of getting a head when flipping a coin. When a die is rolled the sample space is {1,2,3,4,5,6}. From this we can tell that there are 3 out of 6 ways to get an even number from this sample space. The probability for an even number is $\frac{3}{6}$

Step 2

The second step in this process  is to realize that these two events are independent hence we multiply the individual probabilities of the different outcomes to get the odds of a head and an even number. This calculation is shown below,

$P(H\&E)=\frac{1}{2}\times \frac{3}{6}=\frac{3}{12}=\frac{1}{4}.$

The correct answer is D

Number 4

Step 1

The first step is to realize that  the only unique letters in  INNOVATIVE  are {I,N,O,V,A,T,E}, i.e there are only seven unique permutations of these letters.

Step 2

The second step is to calculate the number of 4 permutations of 7 objects.

This is calculated as shown below,

$P(7,4)=\frac{7!}{4!} =\frac{7\times6\times5\times 4!}{4!} =210.$

There are 210 unique permutations of these letters.

Number 5

Step 1

Realize that the number of r combinations of n objects is ,$C(n,r)=^nC_r=\frac{n!}{r!(n-r)!}.$

Step 2

We realize that in this problem we have to make 3 combinations of 11 objects. The calculation to determine the number of combination sis shown below,

$C(11,3)=^{11}C_3=\frac{11!}{3!\cdot(11-3)!} =\frac{11\times 10\times9\times8!}{3!\times 8!}=165$

Number 6

Step 1

We list the outcomes where we first get a head. These outcomes are {HH,HT}. Next we list the outcomes in which we get a tail first. These outcomes are {TH,TT }

Step 2

In this step we combine all the outcomes step 1. The combined list of outcomes is {HH,HT, TH, TT}

Answer 2

The answer for question 2 would be 2/3. I just took the unit test.