Q: Efrain has a collection of antique watches that he likes to display on the mantle above his fireplace. Efrain has 5 different Longines watches, 3 different Zenith watches, 8 different Rolex watches, and 6 different Cartier watches. In how many different ways can Efrain line up these watches along his mantle if Efrain wants watches from the same brand to be kept together?

$$4! \times 5! \times 3! \times 8! \times 6!$$

Q: A standard deck of 52 cards is shuffled and 5 cards are drawn. What is the probability that all 5 cards are from the same suit?

$$\dfrac{4 \times \binom{13}{5}}{\binom{52}{5}}$$

Q: A hospital has 8 nurses and 5 doctors. In how many ways can a team of 3 nurses and 2 doctors be selected?

$$\binom{8}{3} \times \binom{5}{2}$$

Question 1

Let $$U = \{a, b, c, d, e, f, g, h, i, j\}$$ and let $$A = \{b, c, g, h\}. $$How many subsets does $$A^c$$ have?

Accepted Answer

$$2^6$$

Question 2

A group of 48 people went to the Gauss Observatory. It is discovered that 25 of these people visited the Moon Landing exhibit and 30 people visited the Planetarium. If exactly 10 people visited neither of those features, how many people in total visited only the Moon Landing exhibit?

Accepted Answer

8

Question 3

Using the following tree diagram, and the knowledge that $$n(B') = 20$$, determine $$n(A' \cap B).$$

Accepted Answer

7

Question 4

Find the number of different ways to give 40 cents in change using only quarters (25 cents), dimes (10 cents), and/or nickels (5 cents).

Accepted Answer

6

Question 5

Efrain has a collection of antique watches that he likes to display on the mantle above his fireplace. Efrain has 5 different Longines watches, 3 different Zenith watches, 8 different Rolex watches, and 6 different Cartier watches. In how many different ways can Efrain line up these watches along his mantle if Efrain wants watches from the same brand to be kept together?

Accepted Answer

$$4! 	imes 5! 	imes 3! 	imes 8! 	imes 6!$$

Question 6

Western's marching band has two colours of outfits: purple and silver. Thirteen of the band members are wearing the purple outfit and thirteen others are wearing the silver coloured outfit. In how many different ways can these twenty-six people form a line to march and play if no two people wearing the same coloured outfit are allowed to be side-by-side?

Accepted Answer

$$2 	imes 13! 	imes 13!$$

Question 7

Nine different spices are being placed on a rotating, circular spice rack. In how many ways can the spices be arranged on this rack?

Accepted Answer

$$8!$$

Question 8

To celebrate their hockey team's recent victory the fourteen members of the Firehawks are going out to a pizza dinner. They'll all sit down around a large circular table to eat. In how many ways can they be sat down if the five players from their starting lineup all decide to sit together?

Accepted Answer

Question 9

Because of his good grades at school this year, Mitchell's parents have agreed to purchase him four action figures. The store has thirty-two distinct action figures available. In how many ways can Mitchell select four distinct action figures to receive?

Accepted Answer

$$\binom{32}{4}$$

Question 10

A cookie vendor has four different large cookies, each for 2 dollars, and seven different small cookies, each for 1 dollar. Samira intends to buy a total of 3 dollars worth of cookies from the vendor. In how many ways can the cookies be purchased?

Accepted Answer

$$\binom{4}{1} + \binom{7}{2}$$

Question 11

In how many different ways can the letters in the word ADOPTION be arranged if T cannot be the last letter in the arrangement?

Accepted Answer

$$8! - 7!$$

Question 12

Eighty different toys are being sent to the local orphanage. They will be placed in identical boxes to be sent. Each of three boxes will contain exactly twenty toys. In how many different ways can these toys be packed up to be sent to the orphanage?

Accepted Answer

$$\dfrac{80!}{(20!)^3$}$$

Question 13

Eighty different toys are being sent to some of the orphanages in town. They will be placed in different boxes to be sent. Each of these boxes will contain exactly twenty toys. In how many different ways can these toys be packed up and sent if one box is going to the Oakridge Orphanage, one box is destined for the South London Orphanage, and the remaining two boxes will go to the Stoney Creek Orphanage?

Accepted Answer

$$\dfrac{80!}{20!20!20!20!}$$

Question 14

A total of 32 symphony members will play in the Spring concert, while 35 of the members will be at the Fall concert. Three of the members will miss all but the Winter concert. None of the members will miss all the concerts. A total of nine members of the Royal Nunavut Symphony will miss the Winter concert. Only one member will attend the Fall and Winter concerts but not the Spring concert. Finally there are two symphony members who will attend the Spring concert, but cannot play in the other two concerts. How many members of the Royal Nunavut Symphony are able to attend all three concerts this year?

Accepted Answer

1

Question 15

Gunther is getting older and has decided to pass down some family heirlooms to his three children: Abigail, Beatrice, and Craig. Gunther has a total of 13 different heirlooms to pass down. In how many ways can he assign the heirlooms to the children if Craig gets at most 2 heirlooms (and possibly gets none) and Beatrice must receive exactly 4 heirlooms?

Accepted Answer

$$\sum_{k=0}^2 \binom{13}{4} \binom{9}{k} \binom{9-k}{9-k}$$

Question 16

A standard deck of 52 cards is shuffled and 5 cards are drawn. What is the probability that all 5 cards are from the same suit?

Accepted Answer

$$\dfrac{4 	imes \binom{13}{5}}{\binom{52}{5}}$$

Question 17

A committee of 5 is to be formed from 8 men and 7 women. What is the probability that the committee contains at least 3 women?

Accepted Answer

$$\dfrac{\binom{7}{3}\binom{8}{2} + \binom{7}{4}\binom{8}{1} + \binom{7}{5}}{\binom{15}{5}}$$

Question 18

A password consists of 4 letters followed by 2 digits. How many different passwords can be formed if no letter or digit is repeated?

Accepted Answer

$$P(26,4) 	imes P(10,2)$$

Question 19

A hospital has 8 nurses and 5 doctors. In how many ways can a team of 3 nurses and 2 doctors be selected?

Accepted Answer

$$\binom{8}{3} 	imes \binom{5}{2}$$

Combinatorics and Probability: Study Notes for MATH 1228

Guida di studio - Domande di pratica

Study guide - Flashcard