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### Monica Capula
## Counting questions and Mathematical Writing
#### Five original counting questions
1. How many different ways can you rearrange the work "CAT"
Answer:6
How: Summation, writing all combinations, then adding it all up.
Equation/Work:CAT,ATC,TAC,CTA,ACT,TCA
2. There are 4 people at a conference and each shakes everyones hands, how many times was each hand shake? 
Answer:6
How: using the sum principle with diagram. The diagram using vertices as people and edges as handshakes
Equation/Work:
![](https://i.imgur.com/7SEb7Z8.jpg)

3. You are asked to flip a coin three times and to record the outcome for each flip. How many possible outcomes are there?
Answer: 8 outcomes
How: using product principle, there are 3 flips and 2 outcomes.Creating combination and adding them up
Equation/Work:HHH,TTT,HHT,TTH,HTH,HTT,THT,THH

4. You play a lottery, you have to choose five number and each number between 1 and 29, inclusive. How many ways can you fill out the lottery card?
Answer:362880
How:Using product notation
Equation/Work:
20
∏ k =19x18x17x16x15x14x13x12x11x10x9x8x7x6x5x4x3x2x1=3200000
k=1

5. How many ways can 10 cards be organized?
Answer: 3,628,800
How: Product notation, they're 10 cards by using the factorial method, it will tell you how many combiniations. 
Equation/Work:
10
∏ k   =1*2*3*4*5*6*7*8*9*10=3,628,800
k=1