NRP Math Challenge Club
Week 9 : Counting
Multiplicative Rule
1. You have three shirts and four pairs of pants. How many outfits consisting of one short and one
pair of pants can you make? (12)
3 × 4 = 12
2. In how many ways can we form an international commission if we must choose one European
country from among 6 European countries, one Asian country from among 4, one North American
country from among 3, and one African country from among 7? (504)
6 × 4 × 3 × 7 = 504
3. In how many ways can we form a license plate if there are 4 characters, none of which is the letter
O, the first of which is a numerical digit (0-9), the second of which is a letter, and the remaining
two of which can be either a digit or a letter (but not the letter O)? (306,250)
10 × 25 × 35 × 35 = 306250
4. For each of 8 colors, I have one shirt and one tie of that color. How many shirt-and-tie outfits can I
make if I refuse to wear a shirt and a tie of the same color? (56)
8 × 7 = 56
5. Suppose that I have 6 different books, 2 of which are math cooks. In how many ways can I stack
my 6 books on a shelf if I want a math book on both ends of the stack? (48)
2 × 4 × 3 × 2 × 1 × 1 = 48
6. 12 balls numbered 1 through 12 are placed in a bin. In how many ways can 3 balls be drawn, in
order, from the bin, if:
a. each ball remains outside the bin after it is drawn? (1320) 12 × 11 × 10
b. each ball is placed back into the bin immediately after it is drawn? (1728) 𝟏2 × 12 × 12
c. the first ball is replaced after it is drawn but the second ball remains outside the bin? (1584)
12 × 12 × 11
7. On the island of Mumble, the Mumblian alphabet has only 5 letters, and every word in the
Mumblian language has no more than 3 letters in it. How many words are possible? (A word can
use a letter more than once, but 0 letters does not count as a word.) (155)
1 letter = 5 words, 2 letters = 5 × 5 = 25 words, 3 letters = 5 × 5 × 5 = 125 words
5+25+125=155
NRP Math Challenge Club
8. How many 3-letter words can we make from the letters A,B,C,and D, if we are allowed to repeat
letters, and we must use the letter A at least once? (37)
1A: 3 × 3 × 3 = 27, 2As: 3 × 3 = 9, 3As: 1
1+9+27 = 37
Permutations, Combinations
1. You have 5 paintings and only 2 hooks, how many different ways are there to hang the
paintings? (20)
5C2
= 5 × 4=20
2. Simon has 5 paintings that he plans to display on a wall that only has 4 hooks. Nancy has 5
paintings that she plans to display on a wall with 5 hooks. Who has more possible ways to hang
his/her paintings? (same)
Simon : 5C4 = 5 × 4 × 3 × 2 = 120
Nancy : 5C5 =5 × 4 × 3 × 2 × 1 = 120
3. Dasha wants to arrange the seven scrabble letters she has in every possible way so that she
can determine if she has a 7-letter word. How many different ways are there for Dasha to
arrange all seven letters? (5040)
7C7 =
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
4. Ten people are competing for “the Voice”. There will be a 1st place and a 2nd place awarded.
How many different ways can the 1st and 2nd place be awarded? (90)
10C2 =
10 × 9 = 90
5. After a long journey, Michael finally reached the door to the cave that contained the treasure
he had been seeking for over 5 years. The only way to open the door was to place 7 different
colored spherical objects into their correct places on the cave door. If Michael started at 12:00
noon and took 15 seconds to place each different arrangement into the door, at what time
would he complete all possible permutations? (9 am next morning)
7C7 = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
5040× 15 ÷ 60 ÷ 60 = 21 ℎ𝑜𝑢𝑟𝑠
12 noon +21 hours = 9am next morning
6. In a classroom of 5 children, the teacher wants 2 volunteers to go to the office. How many
different groups can be formed from the class of 5 children? (BE CAREFUL, is this the same as
question 1?) (10)
(5× 𝟒) ÷ 𝟐
NRP Math Challenge Club
7. Eight people have volunteered for a secret mission that requires only 3 people. How many
different combinations are possible? (56)
(8 × 7 × 6) ÷ (3 × 2) = 56
8. If 100 people are required to introduce themselves to each other and shake hands with each
person one time, how many different handshakes will take place? (4950)
(100 × 99) ÷ 2 = 4950
9. (*) Mr. Kwong has ten people in his class and wants to make two lab groups of 3 people each,
(the remaining four people just go home). How many ways are there for him to do this? (2100)
10×9×8×7×6×5
6!
= 210
6×5×4
3×2
= 20
210 ×20
2
= 2100
(ways to pick 6 people) (ways to divide 6 people in 2 groups) (each group is counted 2 times)
Counting with restriction
1. In how many ways can I arrange 3 different math books and 5 different history books on my
bookshelf, if I require there to be a math book on both ends? (4320)
3 × 2 × 6! = 4320
2. The Smith family has 4 sons and 3 daughters. In how many ways can they be seated in a row of
7 chairs such that all 3 girls sit next to each other? (720)
Treat the 3 girls as a “supergirl” block, GBBBB, BGBBB, BBGBB, BBBGB,, BBBBG
This can be done in 5! ways, and the girls within the block can be arranged in 3! ways.
5! × 3! = 720
3. The UN has 6 German delegates, 5 French delegates, 3 Italian delegates. In how many ways can
these 14 delegates sit in a row of 14 chairs, if each country’s delegates insist on all sitting next
to each other? (3,110,400)
3! × 6! × 5! × 3!
4. Our math club has 20 members and 3 officers: President, Vice President, and Treasurer.
However, one member, Betty, hates another member, Bob. How many ways can we fill the
offices if Betty refuses to serve as an officer if Bob is also an officer? (6732)
Total ways: 20 × 19 × 18 = 6840
In office together: 3 × 2 × 18 = 108
6840−108 = 6732
NRP Math Challenge Club
In English we use the word "combination" loosely, without thinking if the order of things is
important. In other words:
"My fruit salad is a combination of apples, grapes and bananas" We don't care
what order the fruits are in, they could also be "bananas, grapes and apples" or
"grapes, apples and bananas", its the same fruit salad.
"The combination to the safe is 472". Now we do care about the order. "724"
won't work, nor will "247". It has to be exactly 4-7-2.
So, in Mathematics we use more accurate language:
If the order doesn't matter, it is a Combination.
If the order does matter it is a Permutation.
Permutations
There are basically two types of permutation:


Repetition is Allowed: such as the lock above. It could be "333".
No Repetition: for example, the first three people in a running race. You can't be
first and second.
Combinations
There are also two types of combinations (remember the order does not matter now):


Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)
No Repetition: such as lottery numbers (2,14,15,27,30,33)