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Emma's Dilemma My name is Andrew Donald MacKay; this name has 18 letters and two spaces in it. * 4 A's * 3 D's * 2 N's * 1 R * 1 E * 1 W * 1 O * 1 L * 1 M * 1 C * 1 K * 1 Y * 2 Spaces How many different ways can I write the letters of my name? To investigate this problem I started with shorter names, e.g.) Lucy Jo oj 2 Tim tmi itm imt mit mti 6 Lucy luyc lyuc lycu lcyu lcuy ulcy ulyc uylc uycl ucly ucyl cluy clyu cylu cyul culy cuyl yluc ylcu yulc yucl yclu ycul 24 Jason jasno jaosn jaons janso janos jsaon jsano jsoan jsona jsnao jsnoa joasn joans josan josna jonas jonsa jnaso jnaos jnsao jnsoa jnoas jnosa ajson ajsno ajosn ajons ajnso ajnos asjon asjno asojn asonj asnjo asnoj aojan aojna aoajn aoanj aonja aonaj anjao anjoa anajo anaoj anoja anoaj sjaon sjano sjoan sjona sjnao sjnoa sajon sajno saojn saonj sanaj sanoj sojan sojna soajn soanj sonja sonaj snjao snjoa snajo snaoj snoja snoaj ojsan ojsna ojasn ojans ojnsa ojnas osjan osjna osajn osanj osnaj osnaj oajsn oajns oasjn oasnj oanjs oansj onjsa onjas onsja onsaj onajs onasj njsoa njsao njosa njoas njaso njaos nsjoa nsjao nsoja nsoaj nsajo nsaoj nojsa nojas nosja nosaj noajs noasj najso najos nasjo nasoj naojs naosj 120 Why does 'n!' give the answer? To find out why the formula n! gives the answer we need to divide up the possibilities of the word into the different possibilities of each letter. This is best demonstrated in a tree diagram In the name Jack, there are 4 letters; this means that for the first letter there are 4 different possibilities. The remaining part of the word then has 3 different possibilities (the 3 other letters). For each of these there is only...

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