GMAT Statistics Combinatorics

Author: Brian Galvin

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Language: English

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GMAT Statistics Combinatorics Notes Download Link

Here’s a structured guide on GMAT Statistics & Combinatorics to help you master these concepts efficiently.

GMAT Statistics & Combinatorics Strategy Guide

1. Statistics Concepts for GMAT

Mean (Average)

• Formula: Mean=∑valuesnumber of values\text{Mean} = \frac{\sum \text{values}}{\text{number of values}}Mean=number of values∑values​
• Properties:
  • The mean is sensitive to outliers.
  • The sum of deviations from the mean is always zero.

Median

• The middle value when numbers are arranged in ascending order.
• If there are odd numbers, the median is the middle term.
• If there are even numbers, the median is the average of the two middle numbers.
• Less affected by outliers than the mean.

Mode

• The most frequently occurring number in a set.
• A set may have one mode (unimodal), multiple modes (multimodal), or no mode (if all numbers appear equally).

Range & Standard Deviation

• Range = Largest number – Smallest number
• Standard Deviation (SD):
  • Measures how far numbers are from the mean.
  • Larger SD means more spread; smaller SD means more clustered.
  • If all numbers in a set are increased/decreased by the same value, SD remains the same.
  • If all numbers are multiplied/divided by a constant, SD is multiplied/divided by the absolute value of that constant.

Quartiles & Interquartile Range (IQR)

• Q1 (First Quartile): Median of the lower half of data.
• Q3 (Third Quartile): Median of the upper half of data.
• IQR = Q3 - Q1 (Measures the spread of the middle 50% of the data).

2. Combinatorics Concepts for GMAT

Fundamental Counting Principle

• If one event can happen in m ways and another independent event can happen in n ways, the total number of outcomes is: $$m\times n$$

Permutations (Order Matters!)

• Used when order matters (e.g., rankings, seat arrangements).

• Formula for arranging n distinct objects:

  P(n,r)=n!(n−r)!P(n, r) = \frac{n!}{(n-r)!}P(n,r)=(n−r)!n!​

  Where:

  • n = Total elements
  • r = Chosen elements

• Special case: Arranging n objects:

  $$n!$$

• Arranging with repetition:
  
  If there are n objects where some are identical, use:

  n!k1!×k2!×…\frac{n!}{k_1! \times k_2! \times \dots}k1​!×k2​!×…n!​

  Example: Arranging the word MISSISSIPPI (11 letters: M=1, I=4, S=4, P=2)

  11!1!4!4!2!\frac{11!}{1!4!4!2!}1!4!4!2!11!​

Combinations (Order Doesn’t Matter!)

• Used when order does not matter (e.g., selecting a committee).

• Formula:

  C(n,r)=n!r!(n−r)!C(n, r) = \frac{n!}{r!(n-r)!}C(n,r)=r!(n−r)!n!​

• Example: Choosing 3 out of 5 students:

  C(5,3)=5!3!(5−3)!=5!3!2!=10C(5,3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = 10C(5,3)=3!(5−3)!5!​=3!2!5!​=10

3. Advanced GMAT Combinatorics Strategies

Combination vs. Permutation Shortcut

• If order matters → Use permutation
• If order doesn’t matter → Use combination

"At Least One" Problems

• Use the complement principle: $$\text{P(at least one)}=1-\text{P(none)}$$

Circular Arrangements

• Formula for circular permutations of n objects:

  $$(n-1)!$$

• Example: 6 people sit around a circular table:

  $$(6-1)!=5!=120$$

Combinations with Restrictions

• If a certain person must be included: First select that person, then choose the rest normally.
• If a certain person cannot be included: Exclude that person first, then choose the rest.

4. GMAT Probability & Overlap with Combinatorics

Probability Formula

Probability=Favorable OutcomesTotal Outcomes\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}Probability=Total OutcomesFavorable Outcomes​

"Or" vs. "And" Rules in Probability

• AND (Multiplication Rule, Independent Events): $$P(A\text{ and }B)=P(A)\times P(B)$$
• OR (Addition Rule, Mutually Exclusive Events): $$P(A\text{ or }B)=P(A)+P(B)$$

Binomial Probability (Rarely Tested but Useful)

• If each trial has two outcomes (Success, Failure), use Binomial Distribution: P(k)=C(n,k)pk(1−p)n−kP(k) = C(n, k) p^k (1-p)^{n-k}P(k)=C(n,k)pk(1−p)n−k Where:
  • n = total trials
  • k = number of successes
  • p = probability of success

5. GMAT Test-Taking Strategies for Statistics & Combinatorics

✅ Memorize key formulas (especially permutations, combinations, and probability).

✅ Use logic and elimination—GMAT often provides unnecessary information to distract.

✅ For probability questions, consider using complement method to simplify calculations.

✅ For complex counting problems, break them into smaller, manageable steps.

✅ Use simple cases for verification—If stuck, try smaller numbers to find a pattern.

Would you like additional practice problems or explanations on any topic?

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GMAT Club Math Book

GMAT Statistics Combinatorics

gmat-for-dummies-7th-edition.pdf

Turbocharge your GMAT: Quantitative Question Bank