How many combinations?

Question

Bob has 25 cents in his pocket, but does not have a quarter. If you can tell all his combinations of coins he should have to sum to 25 cents, he will give you the 25 cents. Please list all the combinations and tell me how many there are.

Answer 1

Hey Shelby--you're actually lucky that your teacher wants you to list all the combos, because that's the most fool-proof way to make sure you count every option. (If he/she only wanted to know the number of combos, they'd probably make you solve it with algebra... ugh...)

 

Before you write anything, think about pennies first. He either has to have 0, 5, 10, 15, 20, or 25. If he has any number of pennies in between, he couldn't have exactly 25 cents.

 

When you get to nickels, he can have 0, 1, 2, 3, 4, or 5 nickels. Any more than that is more than 25 cents.

 

And with dimes, he can either have 0, 1, or 2 dimes. (3 or more dimes is more than 25 cents.)

 

Now let's get to work. Make a chart with columns labeled pennies, nickels, dimes.

 

In the first horizontal row, in the DIMES column, put 2. Then finish that row with a 1 in the NICKELS and a 0 in the PENNIES. Then do another row with 2 DIMES, this time 0 NICKELS and 5 PENNIES.

 

Those are the only TWO combos you can do with 2 DIMES.

 

Now start rows with 1 DIME, and find a way to create the other 15 cents with NICKELS and PENNIES. There should be FOUR different combos you can make when you start a row with one dime.

 

Now start a row with 0 dimes, and see how many rows you can create using some number (between 1 and 5) of NICKELS and some number of PENNIES.  There should be FIVE different rows.

 

Now start a row with 0 dimes AND 0 nickels, and see how many rows you can make using just pennies. :-) It's just ONE.

 

How many rows are on your chart? That's the number of combinations you can have.

 

All good? Hope this helped!

 

Matt

Answer 2

Our currency only has dimes, nickels and pennies that are less than a quarter. Let's do this systematically starting with the most amount of dimes.

2 dimes leaves ove 5 cents which is either 1 nickel or 5 pennies. I will write this as follows,

2d 1n  and 2d 5p .

 

So if I only have 1 dime, that leaves me with 15 cents. Now I cannot use another dime for this 15 cents leftover because then I would end up with 2 dimes and that has already been counted. the 15 cents must be with only nickels and pennies. So

1d 3n,  1d 2n 5p,  1d 1n 10p, or 1d 0n 15p.

 

What happens if there are no dimes?

0d 5n,  0d 4n 5p,  0d 3n 10p,  0d 2n 15p,  0d 1n 20p, or 0d 0n 25p.

 

These are all of your possibilities.