SOLUTION: How many words can be formed by using all the letters of the word SCHOOL so that vowels never come together?

First we calculate the number of distinguishable ways in which the letters of
"SCHOOL" can be arranged without regard to whether the two Os come together or not.

If the two O's were distinguishable the number would be 6!.  But since the 2
O's can be placed in 2 of the 6 positions in every arrangement, we must divide
by 2!.  Thus there are 6!/2! = 360 distinguishable arrangements of "SCHOOL".

From that number we must subtract the number of arrangements in which the two
O's come together.  This is all the arrangements of these 5 things:

S,C,H,(OO),L

which is 5!

Answer: 6!/2!-5! = 360/2-120 = 240 

Computer check:

The computer printed 24 rows of 10 each.  So 240 is the correct answer:

 1. SCHOLO SCOHLO SCOHOL SCOLHO SCOLOH SCLOHO SHCOLO SHOCLO SHOCOL SHOLCO
 2. SHOLOC SHLOCO SOCHLO SOCHOL SOCLHO SOCLOH SOCOHL SOCOLH SOHCLO SOHCOL
 3. SOHLCO SOHLOC SOHOCL SOHOLC SOLCHO SOLCOH SOLHCO SOLHOC SOLOCH SOLOHC
 4. SLCOHO SLHOCO SLOCHO SLOCOH SLOHCO SLOHOC CSHOLO CSOHLO CSOHOL CSOLHO
 5. CSOLOH CSLOHO CHSOLO CHOSLO CHOSOL CHOLSO CHOLOS CHLOSO COSHLO COSHOL
 6. COSLHO COSLOH COSOHL COSOLH COHSLO COHSOL COHLSO COHLOS COHOSL COHOLS
 7. COLSHO COLSOH COLHSO COLHOS COLOSH COLOHS CLSOHO CLHOSO CLOSHO CLOSOH
 8. CLOHSO CLOHOS HSCOLO HSOCLO HSOCOL HSOLCO HSOLOC HSLOCO HCSOLO HCOSLO
 9. HCOSOL HCOLSO HCOLOS HCLOSO HOSCLO HOSCOL HOSLCO HOSLOC HOSOCL HOSOLC
10. HOCSLO HOCSOL HOCLSO HOCLOS HOCOSL HOCOLS HOLSCO HOLSOC HOLCSO HOLCOS
11. HOLOSC HOLOCS HLSOCO HLCOSO HLOSCO HLOSOC HLOCSO HLOCOS OSCHLO OSCHOL
12. OSCLHO OSCLOH OSCOHL OSCOLH OSHCLO OSHCOL OSHLCO OSHLOC OSHOCL OSHOLC
13. OSLCHO OSLCOH OSLHCO OSLHOC OSLOCH OSLOHC OSOCHL OSOCLH OSOHCL OSOHLC
14. OSOLCH OSOLHC OCSHLO OCSHOL OCSLHO OCSLOH OCSOHL OCSOLH OCHSLO OCHSOL
15. OCHLSO OCHLOS OCHOSL OCHOLS OCLSHO OCLSOH OCLHSO OCLHOS OCLOSH OCLOHS
16. OCOSHL OCOSLH OCOHSL OCOHLS OCOLSH OCOLHS OHSCLO OHSCOL OHSLCO OHSLOC
17. OHSOCL OHSOLC OHCSLO OHCSOL OHCLSO OHCLOS OHCOSL OHCOLS OHLSCO OHLSOC
18. OHLCSO OHLCOS OHLOSC OHLOCS OHOSCL OHOSLC OHOCSL OHOCLS OHOLSC OHOLCS
19. OLSCHO OLSCOH OLSHCO OLSHOC OLSOCH OLSOHC OLCSHO OLCSOH OLCHSO OLCHOS
20. OLCOSH OLCOHS OLHSCO OLHSOC OLHCSO OLHCOS OLHOSC OLHOCS OLOSCH OLOSHC
21. OLOCSH OLOCHS OLOHSC OLOHCS LSCOHO LSHOCO LSOCHO LSOCOH LSOHCO LSOHOC
22. LCSOHO LCHOSO LCOSHO LCOSOH LCOHSO LCOHOS LHSOCO LHCOSO LHOSCO LHOSOC
23. LHOCSO LHOCOS LOSCHO LOSCOH LOSHCO LOSHOC LOSOCH LOSOHC LOCSHO LOCSOH
24. LOCHSO LOCHOS LOCOSH LOCOHS LOHSCO LOHSOC LOHCSO LOHCOS LOHOSC LOHOCS

Edwin