Combinatorics

1. Five boys and 5 girls were nominated for a homecoming celebration at a local school. How
many ways can a king, a queen, and a court of 2 students be selected from those nominated?
Assume the king must be a boy and the queen a girl. (5C1x5C1x8C2=700)

2. A committee consists of 10 people.

a) How many ways can a subcommittee of 3 people be selected from the committee?
(10C3=120)
b) How many ways can an executive subcommittee consisting of 3 people (chairperson,
treasurer, and secretary) be selected from the committee? (10P3=720)
c) Why are the answers to parts a and b are different? (order matters in b)

3. From a deck of 52 cards, how many 5-card hands can be formed in each case?
a) There are only aces or face cards. (16C5=4,368)
b) There are only numbered cards (no letters) (36C5=376,992)
c) There are 2 clubs and 3 diamonds. (13C2x13C3=22,308)
d) There are at least 4 red cards. (26C4x26C1 + 26C5=454,480)
e) There are exactly 3 fives. (4C3x48C2=4,512)
f) There are exactly 2 Queens and exactly 2 hearts (if Queen of hearts – 3C1x12C1x36C2
or no queen of hearts 3C2x12C2x36C1 = 22680+7128 = 29808)

4. From a deck of 52 cards, the 12 face card are removed. From these face cards, 4 are chosen.
How many combinations that have at least two red cards are possible?
(6C2x6C2+6C3x6C1+6C4 = 360)

5. To play in the Super 7 lottery, you must choose 7 numbers from 1 to 47. To play in the Lotto
649 lottery, you must choose 6 numbers from 1 to 49. To win each jackpot, the numbers chosen
must match the numbers drawn by the lottery corporation.
a) Without doing any calculations, which do you think is more likely, winning the Super 7
jackpot or winning the Lotto 649 jackpot?
b) How many ways are there to match exactly 4 numbers in each lottery?
649: 6C4x43C2 =13,545 Super 7: 7C4x40C3 = 345,800

6. On May 17, 1998, the Powerball Lottery in Oregon had a main jackpot of $195 million U.S.
In this lottery, participants choose 5 numbers from 1 to 49 and 1 number from 1 to 42.
a) How many different ways are there to choose the numbers? (49C5x42C1=80,089,128)
b) What is the probability that all 6 numbers chosen are drawn in the lottery? 1 in 80089128

7. There are 8 boys and 12 girls in a drama club. How many ways can a committee of 5
be selected in each case?
a) There must be at least 2 boys. (20C5 – 12C5 – 12C4x8C1 = 10752)
b) There must be at least 2 girls. (12C2x8C3 + 12C3x8C2 + 12C4x8C1 + 12C5 = 14608)
c) There must be more girls than boys. (12C3x8C2 + 12C4x8C1 + 12C5 = 10912)
8. A multiple choice test has 12 questions, with 5 possible answers for each question.
a) If a student were to guess the answer to each question, how many different ways would
there be to answer the test? (512 = 244,140,625)
b) Some sneaky student stole the solution key and is selling info to several students. Here is
the info: 2 answers are A, 4 answers are B, 3 answers are C, 2 answers are D, and 1 answer is
E. How many different answer keys are possible? (12!/(2!4!3!2! = 831,600)

9. The dial on a combination lock contains marking which represent the number from 0 to 59
inclusive. How many three number combinations are possible if the first number must be a
multiple of 4, the second must be a prime number smaller than 20, and the third number must be
different from both of the first two numbers? (14x8x58=6032)

10. Solve (algebraically):

a) nC2 = 15 (n=6)

b) 100P3 100*99*98

4!(n − 5)!
c) =4 (n=6)
(n − 3)!

11. There are 9 people from Lord Byng running in the Sun Run. If Mr. Cacchioni, Mr. Jack and
Mr. Martin are all in the top 3, and Mrs Lopoke and Mr Tutired finish last and second last
respectively, how many different orders can the 9 runners finish in? Assume there were no ties.
(3!x4!=144)
12.
a) How many words can be formed by using all of the letters in SCRUMTRELESCENT?
(15!/2!2!2!2!3!=1.3621608x1010)
b) How many words can be formed by using all of the letters in RACONTEUR if the word must
start and end with a vowel? (4x7!/2!x3=30240)

c) How many words can be formed by using all of the letters of GOVERNATOR if the two R’s
must be together? (9!/2! = 181440)

13. Mario’s Gelato claims to have the most flavours of gelato in all of Canada. They are able to
make 1521520 different tasting triple scoop cones (of which all 3 scoops must be a different
flavour). How many different flavours does Mario’s have? (chocolate on top of vanilla tastes the
same as vanilla on chocolate). (NC3=1521520, N=210)

14. At the start of the pig race at the PNE last summer, there were 8 numbered stalls. There
were only 6 pigs running in the race though, how many different starting arrangements of the
pigs were possible? (8!/2!=20160)

15. In a student council election, there are 3 candidates for president, 3 for secretary, and 2 for
treasurer. Each student may vote for at least one position. How many ways can a ballot be
marked? (3x3x2 + 3x3 + 3x2 + 3x2 + 3 + 3 + 2 = 47 or 4x4x3 -1)