How many arrangements are there of the letters of the word MATHEMATICS in which all the vowels appear consecutively?

Why does the class go from 14-20 instead of 14-19 like the textbook shows? The textbook provides 7 classes but in your answer, it goes down to 6 classes. What is the reason for this? Thank you :)

What is the definition of one-one and give an example.

Suppose we went to choose 2 colors, without replacement, from the 4 colors red, blue, green, and purple. how many ways can this be done, if the order of the choices is taken into consideration? how many ways can this be done, if the order of the choices is not taken into consideration?