# Confusing Sample Space

A two letter word without repetition is formed from four letters {a, b, c, d}. Make the Sample Space and define event A that first letter is a.

I know that how to solve this question but, I am confused by 'without repetition'.

There are two meanings

1) Without repetition of WORD

2) Without repetition of LETTER

What would be the right option ? Would we include {aa, bb, cc, dd} in the Sample Space ?

• Since there is only one word at a time, I assume they mean repetition of "letter". So $aa$ is out, for instance. – lulu Mar 31 '18 at 13:20

The sample space has $4 \times 3 = 12$ ordered outcomes, three of which have a first.

   ab ac ad
ba    bc bd
ca cb    cd
da db dc


If repetition were allowed, then the missing diagonal elements aa bb cc dd would be included, for a total of $4 \times 4 = 16$ outcomes.

Problems with words like LETTER, STATISTICS, and MISSISSIPPI often ask for the distinguishable arrangements of all the letters. For simpler examples, there $3!/2! = 3$ distinguishable arrangements of the letters in MOM:

MMO MOM OMM


And there are $4!/(2! \cdot 2!) = 6$ distinguishable arrangement of the letters in NOON:

NNOO NONO NOON OONN ONON ONNO