1.

A bag contains 5 green and 7 red jellybeans. How many ways can 5 jellybeans be withdrawn from the bag so that the number of green ones withdrawn will be less than 2?

These combinations are 0 green + 5 red, 1 green + 4 red

choose(5,0)*choose(7,5) + choose(5,1)*choose(7,4)
## [1] 196

2.

A certain congressional committee consists of 14 senators and 13 representatives. How many ways can a subcommittee of 5 be formed if at least 4 of the members must be representatives?

The combinations are 1 senator + 4 representatives, 0 senator + 5 representative.

choose(14,1)*choose(13,4) + choose(14,0)*choose(13,5)
## [1] 11297

3.

If a coin is tossed 5 times, and then a standard six-sided die is rolled 2 times, and finally a group of 3 cards are drawn from a standard deck of 52 cards without replacement, how many different outcomes are possible?

The possible outcome combinations : coins combination * die combinations * cards combinations without replacement

coins_comb<- 2^5
die_comb<-6^2
card_comb<-52 * 51 * 50
coins_comb * die_comb * card_comb
## [1] 152755200

4.

3 cards are drawn from a standard deck without replacement. What is the probability that at least one of the cards drawn is a 3? Express your answer as a fraction or a decimal number rounded to four decimal places.

At least 1 card is 3 can be obtained from 1-pr(cards without 3s) since there are 4 of 3s in the cards, count from 48 cards.

without_3 <- (48/52) * (47/51) * (46/50)
round(1-without_3, digit=4)
## [1] 0.2174

5.

Lorenzo is picking out some movies to rent, and he is primarily interested in documentaries and mysteries. He has narrowed down his selections to 17 documentaries and 14 mysteries.

step1.How many different combinations of 5 movies can he rent?

The combinations are 0 doc + 5 mys, 1 doc + 4 mys, 2 doc + 3 mys, 3 doc + 2 mys, 4 doc + 1 mys, 5 doc + 0 mys

choose(17,0)*choose(14,5)+
choose(17,1)*choose(14,4)+
choose(17,2)*choose(14,3)+
choose(17,3)*choose(14,2)+
choose(17,4)*choose(14,1)+
choose(17,5)*choose(14,0)
## [1] 169911

step2. How many different combinations of 5 movies can he rent if he wants at least one mystery?

The combinations are 0 doc + 5 mys, 1 doc + 4 mys, 2 doc + 3 mys, 3 doc + 2 mys, 4 doc + 1 mys

choose(17,0)*choose(14,5)+
choose(17,1)*choose(14,4)+
choose(17,2)*choose(14,3)+
choose(17,3)*choose(14,2)+
choose(17,4)*choose(14,1)
## [1] 163723

6.

In choosing what music to play at a charity fund raising event, Cory needs to have an equal number of symphonies from Brahms, Haydn, and Mendelssohn. If he is setting up a schedule of the 9 symphonies to be played, and he has 4 Brahms, 104 Haydn, and 17 Mendelssohn symphonies from which to choose, how many different schedules are possible? Express your answer in scientific notation rounding to the hundredths place.

The combinations are 3 from 4 brams * 3 from 104 haydn * 17 mendelssohn * 9!

choose(4,3) * choose(104,3) * choose(17,3) * factorial(9)
## [1] 1.797428e+14

7.

An English teacher needs to pick 13 books to put on his reading list for the next school year, and he needs to plan the order in which they should be read. He has narrowed down his choices to 6 novels, 6 plays, 7 poetry books, and 5 nonfiction books.

Step 1. If he wants to include no more than 4 nonfiction books, how many different reading schedules are possible? Express your answer in scientific notation rounding to the hundredths place.

Maximum choice of nonfiction is 4, The combinations are 0 nonfiction + 13 others, 1 nonfiction + 12 others, 2 nonfiction + 11 others, 3 nonfiction + 10 others, 4 nonfiction + 9 others with orders.

total <- 6+6+7+5
total
## [1] 24
format((choose(24,13)*choose(5,0)+
choose(24,12)*choose(5,1)+
choose(24,11)*choose(5,2)+
choose(24,10)*choose(5,3)+
choose(24,9)*choose(5,4))*factorial(13), digit=3)
## [1] "4.18e+17"

Step 2. If he wants to include all 6 plays, how many different reading schedules are possible? Express your answer in scientific notation rounding to the hundredths place.

The combinations are 6 plays and 7 others, and the totoal is 24.

format(choose(6,6)*choose(24-6,7)*factorial(13), digit =3)
## [1] "1.98e+14"

8.

Zane is planting trees along his driveway, and he has 5 sycamores and 5 cypress trees to plant in one row. What is the probability that he randomly plants the trees so that all 5 sycamores are next to each other and all 5 cypress trees are next to each other? Express your answer as a fraction or a decimal number rounded to four decimal places

The possible scenarios are 5 sycamores are the first half and 5 cypress located at the rest of half, or vice and versa. So, there are two ways to arrange these 10 trees.

round(2/choose(10,5), digit = 4)
## [1] 0.0079

9.

If you draw a queen or lower from a standard deck of cards, I will pay you $4. If not, you pay me $16. (Aces are considered the highest card in the deck.)

Step 1. Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.

There are 11 x 4 cards from 2 to queen, and 2 x 4 cards for king to Ace.

props_2_queen <- (11*4/52) * 4 
props_king_Ace<- (2*4/52) *(-16)
props <- props_2_queen + props_king_Ace
round(props, digit = 2)
## [1] 0.92

Step 2. If you played this game 833 times how much would you expect to win or lose? Round your answer to two decimal places. Losses must be expressed as negative values.

round(833 * props, digit = 2)
## [1] 768.92

would win $ 768.92.