How many different combinations (groups of 1, 2, 3, 4, or 5) can be made from a set of 5 different coins, each with a head side and a tail side?

Question

How many different combinations (groups of 1, 2, 3, 4, or 5) can be made from a set of 5 different coins, each with a head side and a tail side?

 

i.e. a 50cent piece, a quarter, a dime, a nickel, and a penny;  

 

You could have a 50cent piece head with a dime tail, or a penny head with the quarter head, etc.

 

I think it would come out to 234 combinations, but I would like the formula so I could use it for a 4 set combo and a 3 set combo as well.

 

Thanks!

Answer 1

A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected. Since you aren't interested in what order the coins are in, you're looking to solve using the combination formula.

 

The formula for finding a combination looks like this: (ⁿ/_r) = (^n!/_r!(n-r)!)

• n = the total number of objects
• r = number of samples drawn at a time
• ! = factorial (the product of that number and all positive integers less than it)

 

Since you've got 5 different coins with 2 different options each, that means you've technically got a set of 10 total objects.

 

One Coin

n = 10

r = 1

 

(^10!/_1!(10-1)!) = 10

 

Two Coins

n = 10

r = 2

 

(^10!/_2!(10-2)!) = 45

 

Three Coins

n = 10

r = 3

 

(^10!/_3!(10-3)!) = 120

 

Four Coins

n = 10

r = 4

 

(^10!/_4!(10-4)!) = 210

 

Five Coins

n = 10

r = 5

 

(^10!/_5!(10-5)!) = 252

 

Now that you've solved for each coin combination, you'll just need to add all of the final numbers together to find your total.

 

10 + 45 + 120 + 210 + 252 = 637

 

Here's a website with some more explanation and a handy combination calculator: http://stattrek.com/online-calculator/combinations-permutations.aspx