word problem

Question

How many different bracelets can be made from 7 white beads and 2 gray ones?

Answer 1

If we assume that every white bead is alike and both gray beads are alike, and if we assume that the bracelet is a circle (all bracelets are, right?), there are only four possible bracelets.

With combination and permutations, it helps me to write out the possibilities. Let's write W for each white bead and G for each gray bead. Each line below shows a different bracelet sequence.

GGWWWWWWW

 

GWGWWWWWW

 

GWWGWWWWW

 

GWWWGWWWW

 

 

At first glance, it might seem that a lot of combinations are missing, but this is why it's important to think about bracelets being circles. In a circular bracelet, GWWWGWWWW and GWWWWGWWW are actually identical. They're written differently based on which bead you started counting with.

 

This is also why none of the four "begin" with a white bead. That's because for any combo written with a W at the beginning, there's already an equivalent one above. For instance, WWWGWGWWW is actually identical to the second one listed.

 

There's a formula for figuring this out, but I think it's easier and shows more comprehension to write out the possibilities, at least when you're dealing with small enough sets of things.

Answer 2

7 X 2 = 14 different combinations. 

 

Look up the "Basic Counting Principle"