Step 1: We need to identify how many seats there are on the bench. We have 5 names, so 5 seats.

Step 2: When 1 person sits in seat 1, he/she cannot sit in the next set, and so on.

Step 3: Let's work out the math...

Seat 1- 5 people can sit
Seat 2- 4 people can sit
Seat 3- 3 people can sit
Seat 4/5-2/1 people/person can sit 

Total possibilities=\(\displaystyle 5\cdot4\cdot3\cdot2\cdot1=120\). We can also say that \(\displaystyle 120=5!\)

This is a permutation. A permutation is an arrangement of objects in a specific order.

The formula for permutations is:

\(\displaystyle _{n}P_{k} = \frac{n!}{(n-k)!} - n(n-1) (n-2)...(n-k+1)\)

This is written as \(\displaystyle _{n}P_{k}\)

\(\displaystyle _{12}P_{3} = 12\times11\times10 = 1,320\)

There are \(\displaystyle 1,320\) possible groups of 3.

This is a permutation. A permutation is an arrangement of objects in a specific order.

The formula for permutations is:

\(\displaystyle _{n}P_{k} = \frac{n!}{(n-k)!} - n(n-1) (n-2)...(n-k+1)\)

This is written as \(\displaystyle _{n}P_{k}\)

\(\displaystyle _{23}P_3=P(23,3)\) represents the number of permutations of 23 things taken 3 at a time.

\(\displaystyle P(23,3) = 23\times22\times21 = 10,626\) 

\(\displaystyle P(7,5)\) is asking to find the permutation of seven items when you want to choose five. When dealing with permutations, order matters.

A permutation is an arrangement of objects in a specific order.

The formula for permutations is:

\(\displaystyle _{n}P_{k} = \frac{n!}{(n-k)!} - n(n-1) (n-2)...(n-k+1)\)

This is written as 

\(\displaystyle \\_nP_k\\_7P_5=P(7,5) = 7\times6\times5\times4\times3 = 2,520\)

\(\displaystyle P(4,4)\) is asking to find the permutation of four items when you want to choose all four. When dealing with permutations, order matters.

A permutation is an arrangement of objects in a specific order.

The formula for permutations in this case will be,

\(\displaystyle P(4,4) = 4!\) or \(\displaystyle 4\) factorial.

\(\displaystyle P(4,4) = 4\times3\times2\times1 = 24\)

Step 1: A handshake MUST ALWAYS be between TWO people.

Step 2: Break down each person and who they can shake hands with:

Person \(\displaystyle 1\) can shake hands with: \(\displaystyle 2,3,4,5,6,7,8,9,10,11\). 
Person \(\displaystyle 2\) can shake hands with: \(\displaystyle 3,4,5,6,7,8,9,10,11\)
Person \(\displaystyle 3\) can shake hands with: \(\displaystyle 4,5,6,7,8,9,10,11\)
Person \(\displaystyle 4\) can shake hands with: \(\displaystyle 5,6,7,8,9,10,11\)
Person \(\displaystyle 5\) can shake hands with: \(\displaystyle 6,7,8,9,10,11\)
Person \(\displaystyle 6\) can shake hands with: \(\displaystyle 7,8,9,10,11\)
Person \(\displaystyle 7\) can shake hands with: \(\displaystyle 8,9,10,11\)
Person \(\displaystyle 8\) can shake hands with: \(\displaystyle 9,10,11\)
Person \(\displaystyle 9\) can shake hands with: \(\displaystyle 10,11\)
Person \(\displaystyle 10\) can shake hands with: \(\displaystyle 11\)
Person \(\displaystyle 11\) already shook everybody's hand..

Step 3: Count how many handshakes each person can make:

Person \(\displaystyle 1\) shakes hands \(\displaystyle 10\) times.
Person \(\displaystyle 2\) shakes hands \(\displaystyle 9\) times.
Person \(\displaystyle 3\) shakes hands \(\displaystyle 8\) times.
Person \(\displaystyle 4\) shakes hands \(\displaystyle 7\) times.
Person \(\displaystyle 5\) shakes hands \(\displaystyle 6\) times.
Person \(\displaystyle 6\) shakes hands \(\displaystyle 5\) times.
Person \(\displaystyle 7\) shakes hands \(\displaystyle 4\) times.
Person \(\displaystyle 8\) shakes hands \(\displaystyle 3\) times.
Person \(\displaystyle 9\) shakes hands \(\displaystyle 2\) times.
Person \(\displaystyle 10\) shakes hands \(\displaystyle 1\) time.
Person \(\displaystyle 11\) already shook everybody's hand.

Step 4: Add up the number of times each person shook hands:

\(\displaystyle 10+9+8+7+6+5+4+3+2+1=55\)

There were \(\displaystyle 55\) handshakes made between these \(\displaystyle 11\) people.

Step 1: Identify if there are any restrictions to how the numbers can be made...

There are no restrictions, so we can have repeating numbers.

Step 2: Determine how many numbers can go in each slot..

First Slot: 7 choices
Second Slot: 7 Choices
Third Slot: 7 choices

Step 3: Multiply the choices for all three sets together:

\(\displaystyle 7\cdot7\cdot7\equiv7^3\equiv 343\)

We can create \(\displaystyle 343\) different three-digit numbers...

Step 1: See if there are any restrictions..

We see we want only non-repetitive numbers...

Step 2: Find how many numbers can be put in each spot:

First Spot: \(\displaystyle 8\)
Second Spot: \(\displaystyle 7\)
Third Spot: \(\displaystyle 6\)

Step 3: Multiply the number of choices of each spot

\(\displaystyle 8\cdot6\cdot7=336\)

I can create \(\displaystyle 336\) non-repetitive three-digit numbers...

Step 1: Count how many letters are in the word MISSISSIPPI...

There are 11 numbers.

Step 2: Find which letters repeat, and how many times it repeats:

\(\displaystyle I-4\) times
\(\displaystyle S-4\) times
\(\displaystyle P-2\) times

Step 3: Use formula for arranging letters:

\(\displaystyle \frac {(Total\\Letters)!}{(Repeats)!}=\frac {11!}{2!4!4!}\)

Step 4: Expand:

\(\displaystyle \frac {11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{4\cdot 3\cdot 2\cdot 1\cdot 2\cdot 1\cdot 2\cdot 1}\)

Step 5: Simplify Step 4:

\(\displaystyle \frac {11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5}{4}=11\cdot10\cdot9\cdot2\cdot7\cdot6\cdot5=34650\)

I can rearrange the letters in MISSISSIPPI \(\displaystyle 34650\) times...

Step 1: Count how many letters are in the word CORRECT...

There are \(\displaystyle 7\) letters.

Step 2: Find any letters that repeat and how many times they repeat:

C (two times), R (two times)

Step 3: Find how many ways can I arrange the letters: