How many different 3 letter "words" can be made using letters of
the word "MATHEMATICS"?
Case 1: The number of 3-letter "words" from "MATHEICS" (i.e., with
no letters occurring twice).
That's the permutations of 8 letters taken 3 at a time, or
8P3 = 8*7*6 = 336 "words".
Case 2: The number containing 2 like letters.
We can choose the letter to occur twice in 3 ways {M,A,T}.
For each of those choices, we can choose the third letter 7 ways.
For each of those 3*7 choices we can place the third (non-
repeated letter) 3 ways (1st,2nd or 3rd, e.g., {AMM,MAM,MMA} ).
That's 3*7*3 = 63 "words".
Grand total: 336+63 = 399 "words".
Here is a computer generation of all 399 "words", 20 rows of 20 each
except the last row has 1 fewer. (20x20-1 = 400-1 = 399):
AAC AAE AAH AAI AAM AAS AAT ACA ACE ACH ACI ACM ACS ACT AEA AEC AEH AEI AEM AES
AET AHA AHC AHE AHI AHM AHS AHT AIA AIC AIE AIH AIM AIS AIT AMA AMC AME AMH AMI
AMM AMS AMT ASA ASC ASE ASH ASI ASM AST ATA ATC ATE ATH ATI ATM ATS ATT CAA CAE
CAH CAI CAM CAS CAT CEA CEH CEI CEM CES CET CHA CHE CHI CHM CHS CHT CIA CIE CIH
CIM CIS CIT CMA CME CMH CMI CMM CMS CMT CSA CSE CSH CSI CSM CST CTA CTE CTH CTI
CTM CTS CTT EAA EAC EAH EAI EAM EAS EAT ECA ECH ECI ECM ECS ECT EHA EHC EHI EHM
EHS EHT EIA EIC EIH EIM EIS EIT EMA EMC EMH EMI EMM EMS EMT ESA ESC ESH ESI ESM
EST ETA ETC ETH ETI ETM ETS ETT HAA HAC HAE HAI HAM HAS HAT HCA HCE HCI HCM HCS
HCT HEA HEC HEI HEM HES HET HIA HIC HIE HIM HIS HIT HMA HMC HME HMI HMM HMS HMT
HSA HSC HSE HSI HSM HST HTA HTC HTE HTI HTM HTS HTT IAA IAC IAE IAH IAM IAS IAT
ICA ICE ICH ICM ICS ICT IEA IEC IEH IEM IES IET IHA IHC IHE IHM IHS IHT IMA IMC
IME IMH IMM IMS IMT ISA ISC ISE ISH ISM IST ITA ITC ITE ITH ITM ITS ITT MAA MAC
MAE MAH MAI MAM MAS MAT MCA MCE MCH MCI MCM MCS MCT MEA MEC MEH MEI MEM MES MET
MHA MHC MHE MHI MHM MHS MHT MIA MIC MIE MIH MIM MIS MIT MMA MMC MME MMH MMI MMS
MMT MSA MSC MSE MSH MSI MSM MST MTA MTC MTE MTH MTI MTM MTS MTT SAA SAC SAE SAH
SAI SAM SAT SCA SCE SCH SCI SCM SCT SEA SEC SEH SEI SEM SET SHA SHC SHE SHI SHM
SHT SIA SIC SIE SIH SIM SIT SMA SMC SME SMH SMI SMM SMT STA STC STE STH STI STM
STT TAA TAC TAE TAH TAI TAM TAS TAT TCA TCE TCH TCI TCM TCS TCT TEA TEC TEH TEI
TEM TES TET THA THC THE THI THM THS THT TIA TIC TIE TIH TIM TIS TIT TMA TMC TME
TMH TMI TMM TMS TMT TSA TSC TSE TSH TSI TSM TST TTA TTC TTE TTH TTI TTM TTS
Note that some are REAL words, ACE, ACT, AHA, AIM, ASH, ATE, CAM, CAT,
EAT, HAT, HEM, HIM, HIS, HIT, ICE, ITS, MAT, SAT, SEA, SET, SHE, SIT,
TAM, TEA, THE, TIE, and probably some more less used words such as
HIE, TIC, and CHI, ETA (Greek letters). :)
Edwin
