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GMAT Combinatorics & Probability Methods

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

Add/Subtract Odds & Evens

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

Quadratic Templates

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

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