SAMPLE College Math

‭Suppose‬ ‭Emily,‬ ‭who‬ ‭resides‬ ‭in‬‭city‬‭A,‬‭wishes‬‭to‬‭go‬‭to‬‭city‬‭C‬‭via‬‭city‬‭B.‬‭It‬‭is‬‭known‬‭that‬‭there‬‭are‬‭three‬ ‭different‬‭routes‬‭between‬‭city‬‭A‬‭and‬‭B,‬‭and‬‭then‬‭there‬‭are‬‭two‬‭different‬‭routes‬‭between‬‭city‬‭B‬‭and‬‭C.‬‭Now‬ ‭we‬ ‭have‬ ‭to‬ ‭calculate‬ ‭how‬ ‭many‬ ‭options‬ ‭does‬ ‭Emily‬ ‭have?‬ ‭Let‬ ‭us‬ ‭understand‬ ‭this‬ ‭by‬ ‭drawing‬ ‭a‬ ‭tree‬ ‭diagram. Tree diagram refers to the graphical representation of all possible outcomes.‬

‭In‬‭the‬‭above‬‭case,‬‭Emily‬‭has‬‭a‬‭total‬‭of‬‭six‬‭options‬‭to‬‭travel‬‭from‬‭city‬‭A‬‭and‬‭City‬‭C‬‭via‬‭city‬‭B,‬‭that‬‭are‬‭ad,‬ ‭ae,‬‭bf,‬‭bg,‬‭ch,‬‭and‬‭ci.‬‭Hence,‬‭there‬‭are‬‭a‬‭total‬‭of‬‭six‬‭options.‬‭It‬‭will‬‭be‬‭tough‬‭to‬‭count‬‭the‬‭total‬‭possibilities‬ ‭if the sample space is much higher. Sample space refers to the set of all possible outcomes.‬ ‭B. Fundamental Counting Principle‬ ‭If‬‭one‬‭event‬‭occurs‬‭in‬‭m‬‭ways,‬‭and‬‭another‬‭event‬‭occurs‬‭in‬‭n‬‭ways‬‭after‬‭the‬‭occurrence‬‭of‬‭m,‬‭then‬‭the‬ ‭total‬ ‭number‬ ‭of‬ ‭ways‬ ‭in‬ ‭which‬ ‭the‬ ‭two‬ ‭events‬ ‭can‬ ‭occur‬ ‭is‬ ‭calculated‬ ‭as‬ ‭m‬ ‭X‬ ‭n‬ ‭ways.‬ ‭In‬ ‭the‬ ‭above‬ ‭example,‬‭Emily‬‭can‬‭travel‬‭between‬‭city‬‭A‬‭and‬‭city‬‭B‬‭in‬‭three‬‭possible‬‭ways,‬‭and‬‭the‬‭number‬‭of‬‭ways‬‭in‬ ‭which‬ ‭between‬ ‭city‬ ‭B‬ ‭and‬ ‭C‬ ‭is‬‭two.‬‭Applying‬‭the‬‭fundamental‬‭counting‬‭principle,‬‭the‬‭total‬‭numbers‬‭of‬ ‭ways in which Emily can travel are 3 * 2 = 6 ways.‬ ‭Let‬ ‭us‬ ‭take‬ ‭a‬ ‭few‬ ‭more‬ ‭examples‬ ‭to‬ ‭gain‬ ‭expertise‬ ‭in‬ ‭the‬ ‭concept.‬ ‭Remember‬ ‭the‬ ‭Cocoberry‬ ‭Frozen‬ ‭Yogurt‬‭offering‬‭the‬‭choice‬‭of‬‭6‬‭base‬‭flavors‬‭of‬‭yogurt‬‭and‬‭12‬‭options‬‭of‬‭toppings?‬‭We‬‭can‬‭calculate‬‭the‬ ‭total‬‭number‬‭of‬‭possibilities‬‭of‬‭different‬‭yogurts‬‭from‬‭the‬‭store‬‭using‬‭the‬‭counting‬‭principle.‬‭In‬‭this‬‭case,‬ ‭m‬ ‭=‬ ‭6‬ ‭and‬ ‭n‬ ‭=‬ ‭12.‬ ‭Thus,‬ ‭we‬ ‭can‬ ‭say‬ ‭that‬ ‭there‬ ‭are‬ ‭6*12‬ ‭=‬ ‭72‬ ‭ways‬ ‭of‬ ‭selecting‬‭the‬‭frozen‬‭yogurt‬‭at‬ ‭Cocoberry.‬ ‭Now,‬‭let‬‭us‬‭consider‬‭an‬‭example‬‭with‬‭a‬‭bigger‬‭sample‬‭space.‬‭In‬‭a‬‭country,‬‭the‬‭automobiles‬‭bear‬‭license‬ ‭plates‬ ‭with‬ ‭a‬ ‭combination‬ ‭of‬ ‭different‬ ‭letters‬ ‭and‬ ‭digits,‬ ‭such‬ ‭that‬ ‭the‬ ‭initial‬ ‭2‬ ‭positions‬ ‭are‬ ‭filled‬ ‭by‬ ‭letters‬ ‭and‬ ‭then‬ ‭the‬ ‭last‬ ‭three‬ ‭positions‬ ‭are‬ ‭taken‬ ‭by‬ ‭single‬ ‭digits.‬ ‭Now,‬ ‭what‬ ‭is‬ ‭the‬ ‭total‬ ‭number‬ ‭of‬ ‭possibilities of different license plates be made out of this?‬ ‭We‬‭know‬‭that,‬‭there‬‭are‬‭26‬‭letters‬‭(A‬‭to‬‭Z)‬‭and‬‭a‬‭total‬‭of‬‭10‬‭digits‬‭(from‬‭0‬‭to‬‭9).‬‭The‬‭number‬‭of‬‭ways‬‭in‬ ‭which‬ ‭first‬ ‭position‬ ‭of‬ ‭the‬ ‭license‬‭plate‬‭can‬‭be‬‭filled‬‭is‬‭26,‬‭the‬‭second‬‭position‬‭can‬‭also‬‭be‬‭filled‬‭in‬‭26‬ ‭ways,‬‭the‬‭third‬‭in‬‭10,‬‭the‬‭fourth‬‭in‬‭10,‬‭and‬‭the‬‭fifth‬‭position‬‭in‬‭10‬‭ways.‬‭Hence,‬‭the‬‭total‬‭number‬‭of‬‭ways‬ ‭the license plate can be made is:‬

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