## Necklaces

In how many ways can a necklace be made using 5 identical red beads and 2 identical blue beads?
Source: www.mathcircles.org

SOLUTION
Imagine the 7 positions of the beads are represented by the 7 letters $A,B,C,D,E,F,G$. Of these 7 positions we want to select 5 positions to place the red beads. For example, $B,C,E,F,G$.
Once the 5 positions are selected we can place the 5 identical red beads into them in any order. The 2 blue beads go into the remaining 2 positions. In our example the necklace will end up looking like this
$blue\quad red\quad red\qquad blue\quad red\quad red\quad red$
$A\qquad B\qquad C\qquad D\qquad E\qquad F\qquad G$

So how many ways can we select 5 positions from 7 positions?
Answer: $\binom{7}{5}=21$
There are 21 ways to make a necklace with 5 identical red beads and 2 identical blue beads.