Combinatorics + Probability Flashcards
What is the product principle?
n(A and B) = n(A) x n(B)
Where n(A) means the number of ways of making a choice about A.
When would you use the product principle?
When you wish to select one option from A and one option from B, you use the product principle
When would you use the addition principle?
When you wish to select one option from A or one option from B, where A and B are mutually exclusive, you can use the addition principle.
When can’t you use the addition principle?
When the choices for A and B overlap, i.e. odd and prime numbers on a dice. The choices must be mutually exclusive.
In a class there is an award for best mathematician, best sportsperson, and nicest person. People can receive more than one award. In how many ways can the awards be distributed if there are 12 people in the class?
Product principle:
N(A + B +C)
= 12 x 12 x 12
= 1728
An examination has 10 questions in section A and 4 questions in section B. Calculate how many different ways there are to choose questions if you must choose:
a) one question from each section
b) a question from either section A or section B
a) AND => Product principle
10 x 4 = 40
b) OR => Addition principle
10 + 4 = 14
State how many ways:
a) 4 distinguishable toys can be put into 3 distinguishable boxes
b) 3 distinguishable toys can be put into 5 distinguishable boxes
a) 3^4 = 81
b) 5^3 = 125
The number of permutations (arrangements of n distinct objects
n!
A seven digit number is formed by using each of the digits 1 to 7 exactly once. How many such numbers are even?
Pick the final digit to be even (3 ways)
AND
then permute the reamining 6 digits (6! ways)
=> 3 x 6! = 2160
How many permutations of the letters of the word SQUARE start with 3 vowels
Permute the 3 vowels at the beginning (3! ways)
AND
permute the 3 consonants at the end (3! ways)
Number of ways = 3! x 3! = 36
What is the formula for the number of ways of choosing r objects from n objects?
nCr = n!/(r!(n-r)!)
What is the exclusion principle?
When you count what you are not interested in and subtract it from the total
How many 5-digit codes formed by using each of the digits 1 to 5 exactly once do not end in ‘25’?
How could you do this without using the exclusion principle?
Pick the final digit from {1,2,3,4}(4 ways)
AND
permute the remaining 4 digits (4!)
OR pick the final digit as 5 (1 way)
AND pick the penultimate digit from {1,3,4}(3 ways)
AND permute the remaining 3 digits (3! ways)
(4 x 4!) + (1 x 3 x 3!) = 114
How many 5-digit codes formed by using each of the digits 1 to 5 exactly once do not end in ‘25’?
How would you do this using the exclusion principle?
Count permutations of 5 digit codes (5!) ways
then exclude cases where the last two digits are 25 (1 way)
AND
Permute the remaining 3 digits (3! ways)
5! - 1 x 3! = 114
Theo has 8 different jigsaws and 5 different bears. He chooses 4 toys. In how many ways can he make a selection with at least 2 bears?
HINT: use the exclusion principle
Count combinations of 4 toys from 13 (13C4 ways)
then exclude combinations with no bears AND 4 jigsaws (5C0 x 8C4)
OR
one bear AND 3 jigsaws (5C1 x 8C3)
Number of ways = (13C4 - (5C0 x 8C4 + 5C1 x 8C3))
=365