Math Question: How many ways can all the letters of the word PENCIL be arranged if the letter “L” must come first? ?

If each arrangement consists of the letter L, followed by the other 5 letters in some order, then there are 5! = 120 different arrangements possible. 

Think of it as a process of picking the second letter, then the third, and so on, each choice coming from the set of letters that haven't yet been picked. You can see that you'd have 

5 options for the second letter, 

4 for the third, 

3 for the fourth, 

2 for the fifth, and 

1 choice for the sixth letter. So there would be 

5 * 4 * 3 * 2 * 1 = 120 different possible sequences.