## Find all the ways you can use exactly 50 coins to make a change for a dollar 1.00 bill.

### Solution:-

Given that we can use exactly 50 coins whose worth is 1Dollar(100cents)

We have 5 different types of coins

Penny=1 cent

Nickel= 5 cents

Dime=10 cents

Quarter= 25 cents

Half dollar= 50 cents.

Here we observe that the values of the coins of Nickel, Dime, Quarter, Half a dollar are the multiples of 5.

Therefore we have to choose the number of pennies as multiples of 5 only.

Say for example if we are choosing 10 Nickles, which will be 50 cents, we have to choose 50 more pennies to make up 100cents.if we are choosing to say 5 Quarters, which will be 75cents, we have to choose 25 more cents to make up 100cents. Whichever combination of Nickles or Dimes we take, we end up in choosing the pennies which are multiples of 5.

So, if we are starting to choose from a lot of pennies, we must also take them up in the multiples of 5.

Therefore, apparently we must be looking for the quantities of the pennies, like 50, 45,40,35,30 ..taking from the highest number of pennies coins to the lowest.

Before we dive into the problem, let us check the worth in dollars we get when 50 coins of the same type are chosen.

when all pennies are chosen 50 out of 50pennies = 50 cents (0.5 dollars)

when all Nickles are chosen 50 Nickles = 50*5 = 250cents(2.5 dollars)

when all Dimes are chosen 50Dimes = 50*10 = 500cents(5 dollars)

when all Quartersare chosen 50Quarters = 50*25 = 750cents=(7.5dollars)

when all Half of dollar are chosen 50Half of dollar = 50*50 = 2500cents=(25dollars)

From this above information, it is confirmed that we can not select 50 pennies since it would be dollar 0.5 only.

we can not select any other type of coin completely since they exceed Dollar 1.

Let us choose 45pennies, which would make up to 0.45 dollars.

we need to choose 5 more coins to make the total 50 coins.

Therefore we need to get the rest amount 1-0.45=0.55 dollars from the remaining 5 types of coins.

The only possibility to choose is by taking 1 quarter coin, 2-Dime and 2-Nickle coins which add up to 0.55 dollar.

### Therefore one way of choosing to make a 1-dollar change is 45-Penny,1-Quarter,2-Dime,2-Nickle coins

Let us choose 40pennies, which would make up to 0.40dollars

we need to choose 10 more coins to make the total 50 coins.

Therefore we need to get the rest amount 1-0.4=0.6 dollars from the remaining 6 coins.

The only possibility to choose is by taking 2-Dime coins,8-Nickle coins which add up to 0.6 dollars.

### Therefore another way of choosing to make as a 1-dollar change is 40-Penny,2-Dime,8-Nickle coins

Let us choose 35 pennies, which would make up to 0.35.

we need to choose 15 more coins to make the total 50 coins.

Therefore we need to get the rest amount 1-0.35=0.65 dollars from the remaining 15 coins.

we observe that, even if we take 15 coins as nickles, the sum will be 0.75 dollars which is over and above the 0.65 dollars.

therefore taking 35pennies is not a feasible way to choose.

Further taking a reduced number of pennies say 30, which would make up to 0.3 dollars

And we need to choose 20 more coins whose value must be 1-0.3=0.7 dollars.

Even if we choose all 20 coins as Nickles, the amount will be 0.05*20= $1 which is over and about the 0.7 dollars.

Therefore, the number of ways to choose exactly 50 coins to make a change for a dollar 1.00 will be only 2 which are

### 45-Penny,1-Quarter,2-Dime,2-Nickle coins

### 40-Penny,2-Dime,8-Nickle coins

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