factorial

Prove 0! = 1 Let n be a whole number, where n! represents: The product of n and all integers below it through 1. The factorial formula for n is n! = n · (n - 1) · (n - 2) · ... · 3 · 2 · 1 Written in partially expanded form, n! is: n! = n · (n - 1)! Substitute n = 1 into this expression: n! =...

The tip off for this problem is the 2 phrases: circular table arranged Whenever you see these 2 phrases together, the problem is asking for a circular permutation With n = 6: (6 - 1)! 5! 5 x 4 x 3 x 2 x 1 = 120 ways

The formula is nC2 - rC2 + 1 In this problem: n = 8 and r = 3 8C2 = 28 3C2 = 3 Evaluating, we have: 28 - 3 + 1 26

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This is a symbol for a triple factorial. We have n!!! = n * (n - 3) * (n - 6) * ... * 1 Our subtraction of 3 never goes below one. 12!!! = 12 * 9 * 6 * 3

10!= 10 x 9 x 8 x 7 x 6! So we have: 10 x 9 x 8 x 7 x 6!/6! 6! cancels, so we have: 10 x 9 x 8 x 7 5040

Evaluate 6!/3!5! 6!= 6 * 5! 6 * 5!/3!5! 5!'s cancel 6/3! 3! = 6, so we have: 6/6 = 1

How many ways are there to hang 5 paintings on 5 hooks? 5! = 5 x 4 x 3 x 2 x 1 5! = 120

A baseball team has 25 total players consisting of 15 position players and 10 pitchers. How many different ways are there to arrange the batting order of 9 starting players if only one pitcher is used at a time and the pitcher always bats last. (This means that 8 players are taken from the...

An eccentric millionaire has 5 golden hooks from which to hang her expensive artwork. She wants to have enough paintings so she can change the order of the arrangement each day for the next 41 years. (The same five paintings are okay as long as the hanging order is different.) What is the fewest...

One thousand people in. room decide to shake hands with every other person in the room. Instead of one handshake per couple, each person must shake both of the hands of every person in the room with both his right and his left hand. (Tom will use his right hand to shake Dave's right hand and...

If 100 people are required to introduce themselves to each other and shake hands with each person one time, how many handshakes will take place? We want 100 choose 2 since we have 2 people per handshake: 100C2 = 4950

Five players are going to be picked to start a basketball game. If there are 13 players on the team, how many different combinations of 5 starting players can be made? We want 13 combinations of 5 13C5 = 1287

Eight people have volunteered for a secret mission that requires only 3 people. How many different combinations are possible? We want 8 combinations of 3 8C3 = 56

A music camp with 50 students decided to break the students into barbershop quartets to see which combination of four students sounded the best. How many different barbershop quartets can be made with 50 students so that each possible combinations of four is tried? We want 50 combinations of 4...

Twelve friends went to a movie theater. Because the movie was boring, they decided to figure out how many different ways they could sit in the 12 seats. How many different permutations are there for these 12 friends? 12 taken 12 at a time is written as: 12P12 = 479,001,600

Mrs. Lopez gave a homework assignment over summer vacation to read three books from the following list: a) Call of the Wild b) Wuthering Heights c) Death of a Salesman d) The Cartoon Book of Physics How many possible combinations of three books are there in the list of four books? We need to...

A mathematician has 8 favorite paintings and only 6 wall hooks to hang the paintings. How many different ways can she hang the paintings? 8 paintings taken 6 at a time is written as: 8P6 = 20,160

Jay has 5 paintings that he plans to display on a wall that only has 4 books. Nancy has 5 paintings that she plans to display on a wall with 5 hooks. Who has more possible ways to hang his/her paintings? Jay's ways: 5 P 4 = 120 Nancy's ways: 5 P 5 = 120 Therefore, they have the same number of...

Ten people are competing for the title of "Best Singer in the World". There will be a 1st place and a 2nd place awarded. How many different ways can the 1st and 2nd place be awarded?: We have a combinations problem of 10 choose 2. Using our permutations calculator, we see that: 10 P 2 = 90 ways