Question : how many different selections of four letters from the twelve letters of the word REFRIGERATOR contain no R's and two E's?
My attempt:
If there are to be no R's then the selection is limited to
E F I G E A T O
(2 . 1. 6 . 5) = 60 Then 60/2! = 30 selections of 4 letters
I am unsure of whether I am accurate in completely removing the Rs from the selection. Any help appreciated
Since we do not need letter $R$, we need to choose from EFIGEATO. Since we need the selection that consists of $2$ E's, we need to choose from FIAGATO. Note that the positions we need are
EE_ _, E_E_, E_ _E, _EE_, _E_E, _ _ EE
That is the number of arrangements of $2$ same letter E among four positions $=\dbinom{4}{2}=6$
The two blanks are filled by the letters from FIAGATO.
The letter for the first blank can be chosen in $\dbinom{6}{1}=6$ ways
The letter for the second blank can be chosen in $\dbinom{5}{1}=5$ ways.
Therefore, the total number of selections are $6\times6\times5=180$
