From the word ASSASSINATION ,now in how many ways can we form 4 letter word from it ?

Question

My confusion is how to deal with the case when I have 2 same and 2 different ,now since I have 3A's ,4 S so why can't I chose 2A's like 3C2 , what 's the problem in this ?

Answer 1

A S S A S S I N A T I O N

Now, we want 4 letter words from these letters. 

Any 4 letter word is given by ${}^{13} P_4$.

But these include words being repeated like

AAAS
AAAS 
AAAS

as we have given importance to the position of each 'A' but in a word when they come together, the words are the same. 

So, we have to do the hard way. 

3 A's
4 S's
2 I
2 N

Case 1: All four letters are different

${}^6P_4 = 360$ (6 distinct characters)

Case 2: Only 2 letters are same

${}^4C_1 . {}^5C_2 . \frac{4!}{2!} = 480$ (for repetition we can choose any one from A, S, N, I and then permute the 4 with 2 repetitions)

Case 3: 2 letters are repeated twice (like AASS)

${}^4C_2. \frac{4!}{2!.2!}  = 36$

Case 4: Only 3 letters are same

${}^2C_1. {}^5C_1 . 4 = 40$

Case 5: All 4 letters are same

$1$ (only SSSS)

So, in total $360 + 480 + 36 + 40+ 1 = 917$