Example: **Three letters are selected at random from the letters of the word BIOLOGY.**

Find the total number of selections.

The answer is

**not** $\displaystyle \binom{7}{3}$ as you might expect.

Because there are two letters O, you need to find the number of selections with

**1)** no letters O

**2)** one letter O

**3)** two letters O

and then add these together.

**1)** Number of selections with no letter O (ex. B, L and Y)

= number of ways to choose

**three** letters from B, I, L, G, Y

=$\displaystyle \binom{5}{3}$= 10

**2)** Number of selections with one letter O (ex. O, B and L)

= number of ways to choose

**two** letters from B, I, L, G, Y

=$\displaystyle \binom{5}{2}$= 10

**3)** Number of selections with two letters O (ex. O, O and B)

= number of ways to choose

**one** letter from B, I, L, G, Y

= 5

Therefore, total number of selections = 10 + 10 + 5 =

**25**