49 terms

# Order/Counting Principles

GMAT Combinatorics & Probability Methods
###### PLAY
Universal Method for Counting
Slot Method, used in "AND" situations:
1. Create a SLOT for EACH DECISION that you have to make
2. Fill in each SLOT with the NUMBER OF OPTIONS, selecting restricted slots first
3. MULTIPLY
4. (If order DOESN'T matter) DIVIDE by FACTORIAL of numbers of interchangeable elements
When does order matter?
If you scramble the elements & get a different outcome, order DOES matter & use Permutation
Set VS. Code
Set = order DOES NOT matter
Code = order DOES matter
N!
To count/order totals
Fundamental Counting Principle
Total arrangements = Product of # of items
Anagram
When picking groups, total goes in numerator & repeated letters go in the denominator
Multiple arrangements
Multiply the results of multiple anagrams
Arrangements w/Constraints
Total Possible - # of Undesired Arrangements
Glue Method
To find undesired arrangements by making items "stuck together"
Reduce the Pool Method
Ignore preselected moves/items when trying to find the # of possible outcomes
Slot Method
Product of slots completed in order of most restrictive to least restrictive & sometimes involving domino effect situations
Domino Effect
Adjust the outcomes of subsequent events by the outcome of preceding events
"Or" vs "And" scenarios
ADD probabilities for "OR" scenarios &
MULTIPLY probabilities for "AND" scenarios
Speed Tip: Division
Cancel across the fraction line BEFORE performing long division. ALSO, do NOT perform multiplication, if product will be divided. Rather, cancel across the fraction line to simplify.
Simplified Odd/Even +, -, & x Properties
± Same = Even
x at least 1 Even = Even
Odd ± Even = Odd
Odd ± Odd = Even
Even ± Even = Even
Multiply: Odds & Evens
Odd x Even = Even
Even x Even = Even (& divisible by 4)
Odd x Odd = Odd
X
Could be (+), (-), or (0)
Could be (+) or (0)
Sum of a set of Consecutive Integers
Median x # of observations
Counting Consecutive Multiples
For the # of observations:
(((Last - First) +1)/Increment)
*No need to find the first & last multiples.
*Drop everything after the decimal place.

For the Average:
(First + Last)/2
Squared Sum: (x + y)(x + y) = x² + 2xy + y² = (x + y) ²
Squared Difference: (x - y)(x - y) = x² - 2xy + y² = (x - y) ²
Difference of 2 Squares: (x + y)(x - y) = x² - y²
Place Value Question: Occurrence of a Particular Digit in ANY Digit's Place among a set of Consecutive Integers
Look for Patterns. Consider cases for each digit's occurrence in each digit's place & note that as place values increase, the new totals will be a multiple of the previous place values total.
Place Value Question: Occurrence of Particular Digit(s) in a Particular Digit's Place among a set of Consecutive Integers
Firstly, find the difference between the first & last term of the set. Secondly, the next digit's place will often be the smallest increment of recurrence of the pattern & the answer will often be a multiple of that digit's place.
Ordering Places when the setting does not allow
New formula is: (N - 1)!
Ex. Ordering place around a round table
How many TOTAL factors are there of "N?"
The number of factors of "N" will be expressed by the formula (n+1)(o+1)(i+1), where N = aⁿ x b⁰ x cⁱ, and a, b, and c are prime factors of "N" and n, o, and i are their powers.
Similar Figures Ratios
Will have corresponding side lengths in ratio a : b, & their areas will be in ratio a² : b².
Similar Solids Ratios
Will have corresponding sides in ratio a : b, their areas will be in ratio a² : b², & their volumes will be in ratio a³ : b³
Cylinder: Surface Area & Volume
SA = 2πr² + 2πrh
V = πr²h
Exponent Rules
x² (x³) = x(² + ³)
aⁿ(bⁿ) = (ab)ⁿ
x²/x³ = x(² ⁻ ³)
(a/b)ⁿ = aⁿ/bⁿ
(x²)³ = x²(³) = (x³)²
x⁻² = 1/x²
x²/³ = ³√x² = (³√x)²
aⁿ + aⁿ + aⁿ = 3aⁿ

X^3² = X^9
Roots Properties
ⁿ√x/ⁿ√y = ⁿ√(x/y)
ⁿ√x(ⁿ√y) = ⁿ√xy
³√x² = (³√x)² = x²/³
Overlapping Sets
Use Double-Set Matrix (for 2 groups) & Venn Diagram (for 3 or more groups).
General Formula: Total = Group 1 + Group 2 - Both + Neither
Three general properties of GCF and LCM
1. (GCF of m and n) x (LCM of m and n) = m x n
2. The GCF of m and n cannot be larger than the difference between m and n
3. Consecutive multiples of n have a GCF of n
√(-x) =
NOT a real number. Cannot take the square root of a negative number
ⁿ√.5 =
.5ⁿ =
ⁿ√.5 = closer to 1, as n increases
.5ⁿ = closer to 0, as n increases
Relationship: Linear Equations & Variables
When given "x" distinct linear equations & "x" variables, the problem is sufficient
Divisibility Property
If A/B = an integer, then the next number divisible by B, MUST = A + B
Consecutive Integer Factors
Consecutive Integers canNOT share ANY primes
How to manipulate variables in an expression
ISOLATE
X - Y = (+)
Tells you that:
X is larger than Y, that is X is to the right of Y on the number line
X/Y > 1
Tells you that:
1. X & Y have the same sign
2. If X & Y are (+), X > Y
3. If X & Y are (-), X < Y
Probability of (A or B)
P(A or B) = P(A) + P(B) - P(A and B)

*P(A and B) only required when A & B are not mutually exclusive
Simple Interest VS. Compound Interest
Simple Interest = (Principal)(Interest)(Time)
Compound Interest = P(1 + r/n)^nt
Rates & Arithmetic
You can add & subtract individual rates, however to find the average of rates, you MUST first find the total time & total work/distance
How to correct for rounding
For A/B = C, when B is rounded up, increase C
For A/B = C, when B is rounded down, decrease C

For A/B = C, when A is rounded up, decrease C
For A/B = C, when A is rounded down, increase C

For A*B = C, when B is rounded up, decrease C
For A*B = C, when B is rounded down, increase C
Required step on Quant, especially DS Qs
ALWAYS rephrase ALL of the data in the question stem before looking at the answer choices

Making connections between all of the information is essential, as the GMAT does NOT list any extraneous information on the Quant section
Key to Algebraic Translation Qs
Substitute whenever possible to minimize the number of variables created
Inequalities in Q on DS Qs
When trying to prove an inequality, if given a distinct #, no need to prove = sufficient.

However, when given another inequality, must perform calculations to prove
Organization Technique
Think in terms of charts to show relationships for Averages, Rates, Ages, Costs, Change, etc.:

(Average) X (Number) = Total
Black: (24/y) X (y) = 24
White: (24/y+2) X (y+2) = 24