How many ways are there to arrange 9 people in a line? ?

In order to find the answer to this question, lets first answer a simpler on and work our way up.

How many ways can we arrange 2 people in a line?
Let's take a, and b as our people. We can arrange them like ab, ba. 2 ways.

How many ways can we arrange 3 people in a line?
Let's take a, b, and c as our people. We can arrange them like abc, acb, bac, bca, cba, cab. 6 ways.

How many ways can we arrange 4 people in a line?
Let's take a, b, c and d as our people. We can arrange them like abcd, abdc, acbd, acdb, adbc, adcb, bacd, badc, bcad, bcda, bdac, bdca, cabd, cadb, cbad, cbda, cdba, cdab, dabc, dacb, dbac, dbca, dcab, dcba. 24 ways.

From this we can take the rule x = n!, where x is the number of arrangments and n is the number of people. Following this rule we can calculate the answer to be 9! or 362 880 arrangements.