Question

How many different strings can be made from the letters in ORONO, using some or all of the letters?

Answer 1

Task. How many different strings can be made from the letters in ORONO, using some or all of the letters?

Solution. Let $A_{i}$, $i=1,2,3,4,5$, be the number of all $i$-letters words made from some letters in ORONO. We should find

$A=A_{1}+A_{2}+A_{3}+A_{4}+A_{5}.$

Let us write down all possible words.

The list of 1-letters words:

$O,N,R$

so

$A_{1}=3.$

The list of 2-letters words:

$OO,\ \ \ \ \ OR,\ ON,\ RO,\ NO,\ \ \ \ \ \ RN,\ NR,$

so

$A_{2}=7.$

The list of 3-letters words:

$OOO,$

$OOR,\ OON,\ ORO,\ ONO,\ ROO,\ NOO,$

$ORN,\ ONR,\ RON,\ NOR,\ RNO,\ NRO$

so

$A_{3}=13.$

The list of 4-letters words:

$OOOR,\ OOON,\ OORO,\ OONO,\ OROO,\ ONOO,\ ROOO,\ NOOO,$

$OORN,\ OONR,\ ORON,\ ONOR,\ ROON,\ NOOR,\ RONO,\ NORO,\ RNOO,\ NROO,$

so

$A_{4}=18.$

The list of 5-letters words in which N stands before R:

$OOONR,\ OONOR,\ ONOOR,\ NOOOR,$

$OONRO,\ ONORO,\ NOORO,$

$ONROO,\ NOROO,$

$NROOO,$

The list of 5-letters words in which R stands before N:

$OOORN,\ OORON,\ OROON,\ ROOON,$

$OORNO,\ ORONO,\ ROONO,$

$ORNOO,\ RONOO,$

$RNOOO$

and thus

$A_{5}=20.$

Hence

$A=3+7+13+18+20=61.$

Answer. 61.