Essential Question
How important are
combinatorics and probability?
PROBABILITY and COMBINATORICS
Probability is a branch of mathematics that deals
with the likelihood of observing outcomes that can
occur in one event.
Probability in school is introduced through small
experiments – tossing a coin and dice, picking
objects randomly, etc.
Fundamental Principle of Counting and
Combinatorics – permutation and combination are
topics will help to count events more precisely.
Do you know how to count?
Fundamental principle of
counting
Mathematics 10
Mr. joseph b. consignado
Objectives:
1. Count the number of occurrences of an event using a tree
diagram
2. Solve problems using the fundamental principle of counting
Scenario:
You own a small canteen near a school, and you offer affordable combo meals
for the students. A combo meal is a combination of a cup of steamed rice, one
serving of veggie, one serving of meat dish, and a free soup.
Everyday, you prepare a good number of recipes: two veggies, three meat
dishes, and two types of soup.
As part of your marketing strategy, you give “10 + 1” promo. That is, for every
ten combo meals a group of students buy, one extra combo meal is free. The
thrill is that this meal is given by randomly drawing a slip of paper from a box,
wherein the combo meal is written.
What are the possible combinations for this extra combo meal? How many are
the combinations?
Flipping a coin
Flipping a coin twice
Rolling a die
When you roll a die, the possible outcomes are:
Rolling a die, then flip a coin
Fundamental principles of counting
“If one thing can occur in M ways and a second thing can
occur in N ways, and a third thing can occur in P ways, and so
on, the the sequence of things can occur in MxNxPx…..
ways.”
It states that we can find the total number of ways that two
or more separate tasks can happen by multiplying the
number of ways each task can happen separately.
1. Jadine’s Café serves two desserts, cake and pie. They also serve
three beverages, coffee, tea, or juice. Suppose you choose one
dessert and one beverage. How many possible outcomes are there.
2. Manny Pacman is a disc jockey. He chooses different types of
records for each hour of his three-hour program. The possible choices
are listed.
First Hour
Second Hour
Third hour
Rock
Instrumental
Opera
Folk
Jazz
Classical
1. A plate number is made up of three letters from the English alphabet
followed by a three-digit number. How many plate numbers are
possible if
a. The letters and digits can be repeated in the same plate number?
b. The letters and digits cannot be repeated in the same plate number?
2. Three cards are drawn in succession and without replacement from a
deck of 52 cards.
a. Find in how many ways we can obtain the king of hearts, the ace of
diamonds, and the ace of spades in that order.
b. Find the total number of ways in which three cards can be dealt.
3. How many three-digit number can be formed with the digits 2,3,5,6,
and 7 with no repetitions of digit allowed.
3
2
5
6
7
5
6
7
3
6
7
3
5
7
3
5
6
5 digits are given: 2,3,5,6,7
4. Five cards are drawn from a standard deck of cards. Three are black
and two are red.
a. How many possibilities are there?
b. If exactly one of the red cards is a face card. How many possibilities
are there?
5. Mr. Dimagiba’s Quarterly examination has a ten true-false questions.
How many different choices for answering the ten questions are
possible?
6. Four different books are to be arranged on a shelf. How many
arrangements are possible?
7. How many two-digit even numbers can be formed from digits
0,1,2,3,4,5,6,7,8, and 9 if repetition of digits is permitted?
RECITATION
1.
A die and a coin is tossed. Determine the number of different outcomes. List all the
possible outcomes by constructing a tree diagram.
2. A plate number id made up of two consonants followed by a three non zero digits followed
by a vowel. How many plate numbers are possible if
a. The letters and digits cannot be repeated in the same plate number?
b. The letters and digits can be repeated in the same plate number?
RECITATION
3. How many two-digit even numbers can be formed from the digits, 1,2,3,4,5,6,7,8,9 if
repetition of digits in not allowed?
4. In Mathematics class, the teacher presented 10 math trivia's. Each student was given the
option to choose one or two trivia's to answer for additional points under exploratory work. In
how many ways can a student choose the trivia for additional points.
TRY: RECITATION
1. Find the number of possible outcomes for each scenario by drawing a tree diagram
a. A choice of muffin or toast bread with coffee, milk, and juice.
TRY: RECITATION
1. Find the number of possible outcomes for each scenario by drawing a tree diagram
b. Basketball uniform in white, red, blue, yellow, or green which comes in sizes small,
medium, and large.
TRY: RECITATION
2. Six coins are tossed. How many possible outcomes?
3. How many five-digit numbers can be formed using the digits 0,1,2,3,4,5,6,7,8,9 if zero
cannot be the first digit and the given condition for each is to be satisfied.
a. Repetitions are allowed and the number must be even
a. Repetitions are allowed and the number must be divisible by 5.
a. The number must be odd and less than 40000 with repetition allowed.
TRY: RECITATION
4. How many ways can 10 books be arranged on a shelf if one of the books is a bible and it
must be on one end?
ACTIVITY No. 11
1. How many 3 letter could be formed out of 4 letters: A,B,C,D without repetition?
2. A restaurant has 7 varieties of sandwiches, 4 kinds of beverages, and 3 flavors of ice cream. How many
different snacks can be chosen? (Consider a snack to be 1 sandwich, 1 drink, and 1 ice cream)
3. Mauricia has 3 ways to travel from Manila to Batangas, 2 ways from Batangas to Quezon and 6 ways from
Quezon to Bicol. In how many ways can she go from Manila to Bicol?
4. A motor company offers a choice of 4 body styles, 3 motors, and 15 colors. How many car models can be
offered by the motor company?
5. Given four digits 1,2,3, and 4 – digit numbers can you form if digit is repeated?
6. From the set {T, U, V, W, X, Y, Z}, how many four-letter codes can be formed without repeating any letter?
7. How many different numbers can be made from the digits 7,8,9 a. allowing no repetition of the digits
b. repetition in not allowed
8. How many three-digit numbers can be formed from the numbers 1,2,3,4, and 5 if a. repetition is allowed?
b. repetition is not allowed?
9. Daniel has 7 shirts, 4 pants, 10 socks, and 4 pairs of shoes to choose from as he dresses up for a TV
commercial. In how many ways can Daniel dress up?
10. From the digits 1,3,5,7 and 9 and if NO repetition of the digit is allowed, how many numbers can formed
that are:
a. one-digit number
d. four-digit number
b. two-digit number
e. five-digit number
ACTIVITY No. 12
1. A coin tossed four times. Draw a tree diagram to illustrate the possible outcomes.
a. How many different outcomes are possible?
b. How many outcomes have at least 1 head?
c. How many outcomes have at least 2 heads?
d. How many different outcomes have exactly 3 heads?
2. In a student council election, there were 4 candidates for president, 3 candidates for VP, 3
candidates for secretary, and 4 candidates for treasurer. In how many ways can these offices be
filled?
3. New license plates for cars in the Philippines come in 3 letters and 4 digits format.
a. How many license plates in this format are possible?
b. Of these, how many will have all their letters and digit distinct?
4. An organization has to form a core committee consisting of a chairman, vice-chair, and secretary.
These group of officers will be chosen from a group of 8 men and 7 women.
a. In how many can they choose the three officers if there are no restriction in the term of
gender.
b. In how many can they choose the three officers if the chairman must be a male and the
vice chair a female.
Activity No. 13
1. In how many ways can a president and a secretary be chosen from a group of 7 people?
2. An organization has to form a core committee consisting of a chairman, vice-chair, and secretary. These group of
officers will be chosen from a group of 8 men and 7 women. In how many ways can they choose the three officers if
the chairman is to be male or a female and the vice chair and secretary are of the other gender?
3. A box contain 12 cards numbered 1 to 12. A card is drawn from the box. Find the number of ways each of the
following can occur.
a. The card drawn is an odd-numbered card
b. The number on the card is greater than 8 or less than 4.
c. The number on the card is greater or equal to 8.
4. How many 5-letter code(not necessary words with sense) can be formed from the letters A, B, C, D, E, F, G, H, I
and J if
a. there are no restriction?
b. the must begin with a vowel or with a consonant?
c. the word begins with a letter F?
d. the word begins with a letter F followed by a letter E?
5. How many 4-letter code words can be made from letters of the word RELATION, if no letter should be repeated.
6. In how many different ways can a true or false test with 10 questions be answered?
7. How many two-digit odd numbers can be formed the digits 2, 3, 5, 7, 8, 9 if the digits
a. cannot be repeated
b. can be repeated
PERFORMANCE TASK
MATHEMATICS AND TLE
FOOD MENU COMBINATION