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17 September, 16:30

A bus driver pays all tolls, using only nickels and dimes, by throwing one coin at a time into the mechanical toll collector.

a) Find a recurrence relation for the number of different ways the bus driver can pay a toll of n cents (where the order in which the coins are used matters).

b) In how many different ways can the driver pay a toll of 45 cents.

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  1. 17 September, 16:32
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    Answer: a) An = An-1 + An-2

    b) 55ways

    Step-by-step explanation:

    a) a nickel is 5 cents and a dime is 10cent so a multiple of 5 cents is the possible way to pay the tolls in both choices.

    Let An represents the number of possible ways the driver can pay a toll of 5n cents, so that

    An = 5n cents

    Case 1: Using a nickel for payment which is 5 cents, the number of ways given as;

    An-1 = 5 (n-1)

    Case 2: using a dime which is two 5 cents, the number of ways is given as;

    An-2 = 5 (n-2)

    Summing up the number of ways, we have

    An = An-1 + An-2

    From the relation,

    If n = 0, Ao = 1

    n = 1, A1 = 1

    b) 45 cents paid in multiples of 5cents will give us 9 ways (A9)

    From the relation, we have that

    Ao = 1

    A1 = 1

    An = An-1 + An-2

    Ao = 1

    A1 = 1

    A2 = A1+Ao = 1+1 = 2

    A3 = A2 + A1 = 3

    A4 = A3+A2=5

    A5=A4+A3=8

    A6=A5+A4=13

    A7 = A6+A5 = 21

    A8 = A7+A6 = 34

    A9 = A8+A7 = 55

    So there are 55ways to pay 45cents.
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