Permutations: all problems

Evaluate ${}^{6}P_{3}$ and ${}^{7}P_{2}$.

How many 3-digit numbers exist with all odd digits?

How many 3-digit numbers exist with all even digits?

How many 3-digit odd numbers are there with distinct digits?

If you have Alice, Betty, and Chase and you need to choose two of them to be the president and vice president of the math club, how many ways are there to do this?

Find the number of ways of selecting a president, vice president, and secretary from a group of 8 people.

Find the number of 3-digit numbers with all of its digits distinct.

Find the number of ways to arrange 5 different books on a bookshelf.

How many ways are there to pick your favorite and second favorite book amongst 5 books?

There are 10 students at a math contest. How many ways are there to select the first, second, and third place winners?

How many 4-digit numbers are not palindromes? A palindrome is a number that reads the same forward and backward.

A special type of license plate includes 3 distinct letters at the beginning and 4 single-digit numbers after. How many such license plates exist?

There is a school race with 12 students. How many different ways are there to award the gold, silver, and bronze medals if no ties are possible?

How many integers between 1000 and 9999 have four distinct digits?

The Dragonvale Middle School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible?

Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?

How many permutations of the digits 1, 2, 3, 4 have 1 before 3 and 2 before 4? One such permutation is 2134.

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?

Alice, Bob, Claire, Dave, and Emma are sitting around a round table. Alice refuses to sit next to Bob. How many different ways can they sit around the table? (Rotations of the same arrangement are not considered different.)

7 people are lining up for a photograph. If the 2 tallest people must be at the 1st and 7th positions in any order and the shortest person must be in the center, how many ways are there for them to line up?

Five distinct boys and four distinct girls are going to have lunch together around a table. They decide to sit down one by one under the following conditions: no boy will sit down when more boys than girls are already seated, and no girl will sit down when more girls than boys are already seated. How many possible sequences of taking seats exist?