Question

Making change. In Exercise 12, for each i ≥ 0, determine the number of ways c_i of obtaining i¢ if you have the given coins. Ways are distinguished solely by how many of each type of coin they contain.

12. 1 dime, 3 nickels, and 5 pennies.

(i=17; write formula for generating function, but don’t have to simplify it if you don’t need to)

Answer 1

Answer :-

It i easy to count the number of ways if we consider cases

case 1: Using 1 dime

If we use 1 dime (10 cents) then we have to make 7 cents with only 3 nickels and 5 pennies.

The only way is using nickels and pennies.

Hence in this case we have 1 way which is 10+5+1+1

case 2 : using no dimes

sub-case a : Using 3 nickels

In this case there is only one way which adding 2 pennies to obtain 17 cents ( 5+5+5+1+1 )

Sub-case b : Using 2 or less nickels

In this sub case we can use at most 2 nickels and 5 pennies which is a total of 15 cents.

therefore there are no ways to obtain 17 cents.

Thus , the number of ways to obtain 17 cents If we have 1 dime , 3 nickels and 5 pennies is 2

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