Recent questions tagged counting

How many ways are there for eight men and five women to stand in a line so that no two women stand next to each other? [Hint: First position the men and then consider pos...

How many permutations of the letters $\text{ABCDEFGH}$ containthe string $\text{ED}?$the string $\text{CDE}?$the strings $\text{BA and FGH}?$ the strings $\text{AB, DE, a...

How many permutations of the letters $\text{ABCDEFG}$ containthe string $\text{BCD}?$ the string $\text{CFGA}? $the strings $\text{BA and GF}? $the strings $\text{ABC and...

How many bit strings of length $10$ haveexactly three $0s?$ more $0s$ than $1s?$at least seven $1s?$ at least three $1s?$

A coin is flipped $10$ times where each flip comes up either heads or tails. How many possible outcomesare there in total?contain exactly two heads?contain at most three ...

A coin is flipped eight times where each flip comes up either heads or tails. How many possible outcomesare there in total?contain exactly three heads?contain at least th...

How many subsets with more than two elements does a set with $100$ elements have?

How many subsets with an odd number of elements does a set with $10$ elements have?

In how many ways can a set of five letters be selected from the English alphabet?

In how many ways can a set of two positive integers less than $100$ be chosen?

A group contains $n$ men and $n$ women. How many ways are there to arrange these people in a row if the men and women alternate?

How many bit strings of length $12$ containexactly three $1s?$ at most three $1s?$ at least three $1s?$ an equal number of $0s\: \text{and}\: 1s?$

How many bit strings of length $10$ containexactly four $1s?$at most four $1s?$at least four $1s?$an equal number of $0s$ and $1s?$

There are six different candidates for governor of a state. In how many different orders can the names of the candidates be printed on a ballot?

How many possibilities are there for the win, place, and show (first, second, and third) positions in a horse race with $12$ horses if all orders of finish are possible?

In how many different orders can five runners finish a race if no ties are allowed?

Find the number of $5$-permutations of a set with nine elements.

Find the value of each of these quantities.$C(5, 1)$$C(5, 3)$$C(8, 4)$$C(8, 8)$$C(8, 0)$$C(12, 6)$

Find the value of each of these quantities.$P (6, 3)$$P (6, 5)$$P (8, 1)$$P (8, 5)$$P (8, 8)$$P (10, 9)$

Let $S = \{1, 2, 3, 4, 5\}.$List all the $3$-permutations of $S$.List all the $3$-combinations of $S.$

How many permutations of $\{a, b, c, d, e, f, g\}$ end with $a?$

How many different permutations are there of the set $\{a, b, c, d, e, f, g\}?$

List all the permutations of $\{a, b, c\}.$

Let $x$ be an irrational number. Show that for some positive integer $j$ not exceeding the positive integer $n,$ the absolute value of the difference between $j x$ and th...