## Related questions with answers

A country uses as currency coins with values of 1 peso, 2 pesos, 5 pesos, and 10 pesos and bills with values of 5 pesos, 10 pesos, 20 pesos, 50 pesos, and 100 pesos. Find a recurrence relation for the number of ways to pay a bill of n pesos if the order in which the coins and bills are paid matters.

Solutions

VerifiedWhen wish to find a combination of a total peso amount, we will consider all denominations and, in doing so, recursively call the sequence with the peso amount minus the denomination.

A **recursive definition** of a sequence defines the higher terms of the sequence using the terms preceding them.

$(a)$ The following table gives a list of peso coins and bills available.

Coins |
Bills |
---|---|

$1$ | $5$ |

$2$ | $10$ |

$5$ | $20$ |

$10$ | $50$ |

- | $100$ |

So, this makes $4 (\text{coins})+5(\text{bills})=9$ types of payment options.

Suppose $a_n$ denotes the number of ways to pay a bill of $n$ pesos using the given payment options such that the order matters. The given scenario can be broken into **nine** distinct cases depending on which of the nine options is used first. This is expressed in the following table.

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