Counting and Combinatorics - GRE Quantitative Reasoning
Card 1 of 11
State the formula for the number of permutations of $n$ distinct objects taken $r$ at a time.
State the formula for the number of permutations of $n$ distinct objects taken $r$ at a time.
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$P(n,r)=\frac{n!}{(n-r)!}$. Calculates the number of ways to arrange $r$ distinct items from $n$, accounting for order by multiplying descending factorials.
$P(n,r)=\frac{n!}{(n-r)!}$. Calculates the number of ways to arrange $r$ distinct items from $n$, accounting for order by multiplying descending factorials.
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State the formula for the number of combinations of $n$ distinct objects taken $r$ at a time.
State the formula for the number of combinations of $n$ distinct objects taken $r$ at a time.
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$\binom{n}{r}=\frac{n!}{r!(n-r)!}$. Determines the number of ways to select $r$ items from $n$ without regard to order, dividing by $r!$ to eliminate permutations.
$\binom{n}{r}=\frac{n!}{r!(n-r)!}$. Determines the number of ways to select $r$ items from $n$ without regard to order, dividing by $r!$ to eliminate permutations.
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State the multiplication (fundamental counting) principle for $k$ sequential choices with $n_1,\dots,n_k$ options.
State the multiplication (fundamental counting) principle for $k$ sequential choices with $n_1,\dots,n_k$ options.
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$n_1\times n_2\times\cdots\times n_k$ outcomes. Multiplies the options at each independent stage to count total outcomes in sequential decisions.
$n_1\times n_2\times\cdots\times n_k$ outcomes. Multiplies the options at each independent stage to count total outcomes in sequential decisions.
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What is the value of $0!$?
What is the value of $0!$?
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$0!=1$. Defined by convention to ensure consistency in formulas like binomial coefficients and permutations for $r=0$.
$0!=1$. Defined by convention to ensure consistency in formulas like binomial coefficients and permutations for $r=0$.
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What is the factorial definition of $n!$ for an integer $n\ge 1$?
What is the factorial definition of $n!$ for an integer $n\ge 1$?
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$n!=n(n-1)(n-2)\cdots 2\cdot 1$. Represents the product of all positive integers up to $n$, fundamental for counting permutations and combinations.
$n!=n(n-1)(n-2)\cdots 2\cdot 1$. Represents the product of all positive integers up to $n$, fundamental for counting permutations and combinations.
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State the number of distinct permutations of a multiset with counts $n_1,n_2,\dots,n_k$ totaling $n$.
State the number of distinct permutations of a multiset with counts $n_1,n_2,\dots,n_k$ totaling $n$.
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$\frac{n!}{n_1!n_2!\cdots n_k!}$. Adjusts total permutations by dividing by factorials of identical item counts to avoid overcounting indistinguishable arrangements.
$\frac{n!}{n_1!n_2!\cdots n_k!}$. Adjusts total permutations by dividing by factorials of identical item counts to avoid overcounting indistinguishable arrangements.
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State the formula for permutations of $n$ objects when repetition is allowed and order matters, length $r$.
State the formula for permutations of $n$ objects when repetition is allowed and order matters, length $r$.
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$n^r$. Allows each of $r$ positions to independently choose from $n$ options, permitting repetitions.
$n^r$. Allows each of $r$ positions to independently choose from $n$ options, permitting repetitions.
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State the inclusion-exclusion formula for two sets $A$ and $B$.
State the inclusion-exclusion formula for two sets $A$ and $B$.
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$|A\cup B|=|A|+|B|-|A\cap B|$. Adds individual set sizes and subtracts their intersection to correct for double-counting.
$|A\cup B|=|A|+|B|-|A\cap B|$. Adds individual set sizes and subtracts their intersection to correct for double-counting.
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Identify the number of ways to choose a nonempty subset from $n$ elements.
Identify the number of ways to choose a nonempty subset from $n$ elements.
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$2^n-1$. Subtracts the empty set from the total number of subsets of an $n$-element set.
$2^n-1$. Subtracts the empty set from the total number of subsets of an $n$-element set.
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Identify the number of subsets of a set with $n$ elements.
Identify the number of subsets of a set with $n$ elements.
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$2^n$. Arises from each of $n$ elements having two choices: inclusion or exclusion in the subset.
$2^n$. Arises from each of $n$ elements having two choices: inclusion or exclusion in the subset.
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What is the symmetry identity for binomial coefficients $\binom{n}{r}$?
What is the symmetry identity for binomial coefficients $\binom{n}{r}$?
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$\binom{n}{r}=\binom{n}{n-r}$. Reflects that selecting $r$ items to include equals selecting $n-r$ to exclude from the set.
$\binom{n}{r}=\binom{n}{n-r}$. Reflects that selecting $r$ items to include equals selecting $n-r$ to exclude from the set.
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