GRE Quantitative Flashcards: Counting Combinatorics

What this deck covers

This deck focuses on Counting Combinatorics, giving you a quick way to review the definitions, rules, and examples that matter most for GRE Quantitative.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the value of $0!$?

Answer: $0!=1$. Defined by convention to ensure consistency in formulas like binomial coefficients and permutations for $r=0$.

Flashcard 2: What is the symmetry identity for binomial coefficients $\binom{n}{r}$?

Answer: $\binom{n}{r}=\binom{n}{n-r}$. Reflects that selecting $r$ items to include equals selecting $n-r$ to exclude from the set.

Flashcard 3: Identify the number of ways to choose a nonempty subset from $n$ elements.

Answer: $2^n-1$. Subtracts the empty set from the total number of subsets of an $n$-element set.

Flashcard 4: Identify the number of subsets of a set with $n$ elements.

Answer: $2^n$. Arises from each of $n$ elements having two choices: inclusion or exclusion in the subset.

Flashcard 5: State the formula for permutations of $n$ objects when repetition is allowed and order matters, length $r$.

Answer: $n^r$. Allows each of $r$ positions to independently choose from $n$ options, permitting repetitions.

Flashcard 6: State the formula for the number of permutations of $n$ distinct objects taken $r$ at a time.

Answer: $P(n,r)=\frac{n!}{(n-r)!}$. Calculates the number of ways to arrange $r$ distinct items from $n$, accounting for order by multiplying descending factorials.

Flashcard 7: State the inclusion-exclusion formula for two sets $A$ and $B$.

Answer: $|A\cup B|=|A|+|B|-|A\cap B|$. Adds individual set sizes and subtracts their intersection to correct for double-counting.

Flashcard 8: State the multiplication (fundamental counting) principle for $k$ sequential choices with $n_1,\dots,n_k$ options.

Answer: $n_1\times n_2\times\cdots\times n_k$ outcomes. Multiplies the options at each independent stage to count total outcomes in sequential decisions.

Flashcard 9: What is the factorial definition of $n!$ for an integer $n\ge 1$?

Answer: $n!=n(n-1)(n-2)\cdots 2\cdot 1$. Represents the product of all positive integers up to $n$, fundamental for counting permutations and combinations.

Flashcard 10: State the formula for the number of combinations of $n$ distinct objects taken $r$ at a time.

Answer: $\binom{n}{r}=\frac{n!}{r!(n-r)!}$. Determines the number of ways to select $r$ items from $n$ without regard to order, dividing by $r!$ to eliminate permutations.

Flashcard 11: State the number of distinct permutations of a multiset with counts $n_1,n_2,\dots,n_k$ totaling $n$.

Answer: $\frac{n!}{n_1!n_2!\cdots n_k!}$. Adjusts total permutations by dividing by factorials of identical item counts to avoid overcounting indistinguishable arrangements.