Abus 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.

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.

ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga

step-by-step explanation:

idk the answer but just put a

step-by-step explanation: