Counting and Combinatorics - Quantitative Reasoning

1. What is the value of $0!$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Study these flashcards