HPC Ch2 practice

(Calculator) Seniortis is an afflication that commonly strikes [math]12^{\text{th}}[/math] grade students at this time of year. Mr. Hua has devaloped a test that can help detect seniortis. If a student has seniortis, the test will come back positive 83.2% of the time. Conversely, when a student does not have seniortis, the test gives a false positive 11.6% of the time. After testing all 200 [math]12^{th}[/math] grades students, Mr. Hua found that 67.5% of the tests returned positive results. How many of the 200 [math]12^{\text{th}}[/math] grade students really have seniortis?

Mrs. Rowe recently conducted a poll of all 209 [math]8^{\text{th}}[/math] graders at TAS. She discovered that 84 students participate in MUN and 139 students are in advanced algebra 1. The number of students that are both in advanced algebra 1 and participate in MUN are three times the number that are not in either. How many students are in advanced algebra 1 and participate in MUN?

In how many different ways can the letters of the word “OPTICAL” be rearranged so the vowels always come together?

In how many different ways can the letters of the word “CORPORATION” be rearranged so the vowels always come together?

Louis, Ryan, Wish, Max, Joseph, and Chris are going on a road trip to Kenting. Of the group, only Louis, Ryan and Wish have driver’s license and can drive. They plan on renting one minivan that can seat all of them or two cars (each can seat 4 people including the driver). How many different ways can the group arrange themselves for the trip? (Other than the driver, where each person sits does not matter)

The grid shown represents the street of a city.

Determine how many routes from A (Karen’s house) to B (Anna’s house) are possible if Karen only travels south or east?

(Calculator) Jasmine, Jessica, Joseph, Jonathan, Charles, Courtney, Derek and Daniel are lining up to purchase tickets to the middle school musical Oliver. How many different arrangements are possible when they line up if:

(a) Jasmine has to be next to Jessica, but not next to Courtney

(b) Charles does not want to be next to a girl (Jasmine and Jessica are girls, the others are boys)

The grid shown represent the street of downtown Taipei. Determine how many routes from A (Henri’s house) to B (Baltazer’s house) are possible if Henri only travels south or east.

Annabelle, Bryce, Caitlin, Dorothy, Ethan, and Francis are to be seated along one side of a rectangular table. How many different seating arrangements are possible if:

(a) There are no restrictions?

(b) The boys cannot sit next to each other?

(c) Annabelle cannot sit next to Caitlin, and the boys must sit together

The grid shown represent the street of downtown Taipei. Determine how many routes from A (Alvin’s house) to B (Rem’s house) are possible if Henri only travels south or east.

(Calculator) The grid shown represent the streets of downtown Taipei. Determine how many routes from A(Williams’ house) to B(Justin’s house) are possible if William only travels south or east.

Annabelle, Claire, Emily, Joseph, Louis, Max, and Ryan are lining up to purchase tickets to the middle school musical The wizard of Oz. How many different arrangements possible when they line up if:

(a) There are no restrictions

(b) The girls have to be together

(c) Joseph has to be next to Emily

How many triangles can you make using 6 non collinear points on a plane?

Consider a standard deck of 52 cards. How many different 5 card hands can be formed containing exactly 3 of the kind?

From a group of 7 men and 6 women, five people are to be selected to form a commitee. How many ways can this be down if there are at least 3 men on the committee?

In a group of 6 boys and 4 girls, four children are to be selected. In how many differnet ways can they be selected such that at least one boy is selected?

(Calculator) Richard has a standard deck of playing cards. He wants to know how many different hands are possible with the following properties

(a) A straight (5 cards of sequential rank i.e. 2,3,4,5,6)

(b) A straight but all Kings are taken out of the deck

(Calculator) You are the current commissioner of the Taiwan Lottery and you are considering making a small change. The current game has the player select 6 numbers out of a pool of 38 and then select one number out of a poor of 8. In order the increase the jackpot, you are tasked with increasing to total number of possible outcomes. Consider the following options where only one number is changed:
(a) Select 7 numbers out of a pool of 38 and then select one number out of a pool 8
(b) Select 6 numbers out of a pool of 39 and then select one number out of a pool 8
(c) Select 6 numbers out of a pool of 38 and then select one number out of a pool 9
Which one of these options will result in the most outcomes?

(Calculator) Use the Binomial Theorem to expand [math](4+3x)^5[/math].

Find the [math]x^2[/math] term in [math](2+x)^4(1+\frac{1}{x^2})[/math].

Find the [math]x^4[/math] term in [math]\left(3x^2-\dfrac{2}{x}\right)^5[/math].

(Calculator) Determine the coefficient of the [math]x^{33}[/math] in the expansion of [math]\left(2x^3+\dfrac{x}{2}\right)^{17}[/math].

(Calculator) Determine the [math]x^{32}[/math] term in the expansion of [math]\left(2x^3+\dfrac{x}{2}\right)^{16}[/math].

(Calculator) Find the constant term in the expansion [math]\left(3x+\dfrac{2}{x^2}\right)^9[/math].

Identify the scenario as a combination or permutation. If the scenario is combination, type “A”; if the scenario is permutation, type “B”. Otherwise, type “C”.
(a) How many ways can you choose a captain and assistant captain on an 18 player team?

(b) In gym class, you are playing “Follow the Leader”. How many ways can you select 3 leaders from a class of 20?

(c) I have a pen. I have an apple. I have a pen. I have a pineapple. How many unique arrangements of these four objects is possible, i.e. pen-pineapple-apple-pen? Treat the pens as identical.

(Calculator) How many license plates of 4 symbols (letters and digits) can be made using at least two numbers in each if the symbols cannot repeat?

Mr. Lopez wakes up each morning stressing about what to wear. He has 3 different dress shoes, 12 dress shirts and 7 dress pants. He likes to impress his students and never wear the same outfit. How many weeks can he go until the students notice he’s worn the same outfit.

How many even four-digit numbers can be made from the digits 0, 2, 3, 4, 5, or 7 if no digit can be repeated?

(Calculator) Five different mathematics books are to be arranged on Ms. Noda’s desk. How many arrangements are possible?

(Calculator) How many different words (real or imaginary) can be formed using all the letters in the word REARRANGE?

(Calculator) In a dog show, there are three Pomeranians, five Golden Retrievers, two Great Pyrenees, and six English Terriers. If the dogs of the same breed are considered identical, in how many ways can the dogs be lined up in front of the judges?

(Calculator) Twelve mice run in a race. How many different ways can they come in first, second, and third places? Assume that there are no ties.

Evaluate each expression.

(a) [math]C^8_3[/math]

(b) [math]P^9_4[/math]

(c) [math]C^8_2[/math]

How many ways can 6 people work at 3 different classrooms if 2 people are in each classroom and 2 individuals refuse to work together?

(Calculator) Committee of 4 is chosen from 7 men and 6 women. How many different committees can be chosen if :

(a) there are no restrictions

(b) there must be 2 of each gender

(c) at least one of each gender is needed?

(Calculator) In how many ways can 5 boys and 4 girls be arranged on a bench if

(a) The boys and girls are in separate groups? (So a girl would not sit between 2 boys.)

(b) Anne and Jim wish to sit together?

The expansion of [math](x^2+x^{-3})^4[/math] has no

(Calculator) Find the constant term of [math]\left(2a-\dfrac{3}{a^2}\right)^9[/math].