Arrangements of the letters in 'CANADIAN'

[Clifford]

Consider all the possible arrangements of all the letters in the word 'CANADIAN'

A) How many arrangements begin with the letter A?
B) How many arrangements begin with two A's?
C) How many arrangements begin with just 1 A?
D) How many arrangements begin with just 2 A's?

I can't figure out how to calculate how many arrangements with begin or end with a certain letter or a group of letter. If anybody could show me how to do part A, hopefully I can complete the rest of the parts.

Thanks

 

[stapel]

I am assuming that the different copies of the same letter (such as A) are indistinguishable.

A) This reduces to the number of ways to arrange the letters in "CNADIAN". Apply the formula.

B) This reduces to the number of ways to arrange the letters in "CNDIAN". Apply the formula.

C) This differs from (A) because it specifies that the second letter can NOT be A. So the first slot is A; there is only one way to fill that in. Then how many other (non-A) letters do you have? This fills in the second slot. Then the other slots can be any of the remaining letters. I would proceed by cases. For instance, if the second letter is C, then the remaining slots will be filled with the arrangements of the letters in "NADIAN". And so forth.

D) This differs from (B) in the same way that (C) differed from (A). Follow the same sort of reasoning.

Eliz.