Counting

Basics:

• Be careful of the boundary conditions

• Try to come up with a general rule

Example #1

How many integers are less than 600 but greater than 500?

• Be careful of the boundary, not to include 500 nor 600.

• Answer: 599 – 500 = 99

Example #2Jim has a yard that is 60 yard long and 30 yard wide. If he places a host at each corner and places other posts three yards apart along the edge. How many posts are placed surrounding the yard?

• Be careful not to over count corner posts

• Set the pattern: each line with one corner

1*3, 2*3, …10x3; total 10 posts

10 posts

1*3, 2*3, …, 20*3; total 20 posts

20 posts

• Answer: 20 + 10 + 20 + 10 = 60 posts

Venn Diagrams

• Useful when counting among categories that may have overlapping entries

• Working from inside-out helps

Example #3At a party gathering, 19 person have a brother, 15 person have a sister, 7 person have both a brother and a sister, and 6 person don't have any siblings at all. How many person are at the party?

• Recognizing there are 2 categories: having a brother, having a sister.

• Draw the Venn diagrams accordingly.

• Answer: 7 + 12 + 8 + 6 = 33

Only have a brother:19 – 7 = 12

Only havea sister:15 – 7 = 8

Have no brother nor sister: 6

Have bothbrother &sister: 7

Example #4Every student in a classroom has at least one pet. 30 have a cat, 28 have a dog, and 26 have fish. If 13 students have fist and a cat, 15 students have fish and a dog, 11 students have both a cat and a dog, and 4 students have a cat, a dog, and fish. How many students in the classroom?

• Recognizing there are 3 categories: have a cat, have a dog, or have fish. • Draw the Venn diagrams accordingly.

• Answer: 4 + 7 + 9+ 11 + 8+ 8 + 2 = 49

4

11 – 4 = 7

13 – 4 = 915 – 4 = 11

Dog only:30–7–4–11= 8

Cat only:28-7-4-9 = 8

Fish only:26 –4–9–11 = 2

Example #5How many of the smallest 1000 positive integers are divisible by 5, 6, or 7?

• Recognizing there are 3 categories: div by 5, div by 6, div by 7.

• Draw the Venn diagrams accordingly.

Answer: (div-by-5: 200) + (div-by-6: 166)+ (div-by-7: 142) - (div-by-5&6: 33) - (div-by-5&7: 28) - (div-by-6&7: 23) + (div-by-5&6&7: 4)

4

Div by5&6: 33

div by6&7: 23

div by5&7: 28

Div by 5 onlyDiv by 6 only

Div by 7 only

= 200 + 166 + 142 – 33 – 28 – 23 + 4 = 428

Bowling pins and handshakes

• How many pins in the diagram?

Answer: 1 + 2 + 3 + 4 = 10

• What about: 1 + 2 + 3 + … + 100 = ?

Example #6: 1 + 2 + 3 + … + 100 = ?

Assume S = 1 + 2 + 3 + … + 100

we can also write S = 100 + 99 + 98 + … + 1

pair them as shown in circles, we got:

2 * S = 101 * 100 Thus S = 101 * 1000 / 2 = 5050

Fundamental counting principle

If there are m ways that one event can happen, and n ways a second event can happen, then there are m*n ways that both events can happen.

Example #7How many 10 digit whole numbers use only the digits 1 and 0?

• Note that the first digit can’t be 0, and all other digits have 2 choices.

• Answer: 1 * 29 = 512

Example #8How many squares can be formed by 4 of the dots in the unit grid as vertices?

• Keeping organized.

• Recognize all possible category

Example #8

• # of unit square: 9• # of 2x2 square: 4• # of 3x3 square: 1• # of 2x2 diagonal square: 2

• Answer: 9 + 4 + 1 + 2 + 4 = 20• # of 1x1 diagonal square: 4

Factorials and permutations

n ! = n * (n – 1) * (n – 2 ) … * 2 * 1

Example:

How many different "words" can be formed by re-arranging the letters in the word "COUNT"?

Answer: 5! = 5 * 4 * 3 * 2 * 1 = 120

Example # 9

How many different 8 letter "words" can be formed by re-arranging the letters in the word "GEEEETRY"?

• Think of E1 and E2 as two different characters first, we got total # of words: 8!

• Then remove the duplicates, we get the answer: 8! / 2 = 20160

Permutation with restrictions

How many even five digit numbers contains each of the digits 1 through 5?

Working from the last digit, we got:

answer = 2 * 4 * 3 * 2 * 1 = 48

Combinations

• Permutation where the sequence of the elements doesn’t count.

• Can be calculated by removing the repeated ones from the permutation result.

Combination = (permutation of the whole set) / (permutation of the selected set)

Example #10

Remy wants to drink 3 different sodas from a list of 8 sodas. How many different soda combinations he can choose to drink?

• Permutation of the whole set: 8 * 7 * 6

• Permutation of selected set: 3 * 2 * 1

• # of combinations: (8 * 7 * 6) / (3 * 2 * 1)

• Answer: 56

Example #11Tracing the lines starting from A on the unit grid below, how many distinct 7-unit paths are there from A to B?

Example #11

Must move 3 Up-moves in seven moves, and 4 Right moves in seven moves. Thus the question becomes: How many ways to put 4 Rs in 7 slots.

Ways to put 4 Rs in 7 slots: 7 * 6 * 5 * 4 ; divide by the repeated ones: 4 * 3 * 2 * 1, we got the answer: 7*6*5*4 / (4*3*2*1) = 35

Complementary counting

Total # of desired = total # - the # that we don’t want

(Hint: use this strategy if the desired set is a union of different sets)

Example #11

Paul flits a fair coin eight times. In how many ways can he flip at least two heads?

Total count: 28; Count for no head: 1Count for 1 head: 8

Use complementary counting, the answer is: 28 – 1 – 8 = 247