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

### 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

### "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.

### 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

### 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

### 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

### Relationship: Linear Equations & Variables

When given "x" distinct linear equations & "x" variables, the problem is sufficient

### 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