# How many ways can you choose a set of 9 pencils from a selection of 10?

Jul 30, 2017

See a solution process below:

#### Explanation:

There are 10 choices for the first pencil.

There are 9 choices left for the second pencil

There are 8 choices left for the third pencil

and so on

Until the are 2 choices left for the ninth pencil

This means the number of ways you can choose 9 pencils from a selection of 10 is:

$9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 362 , 880$

Jul 30, 2017

$10$ ways

#### Explanation:

When we are asked to "choose" a number of objects from a larger set, we use combinations. This is because the order of the objects does not matter. If the order did matter, we would use permutations.

In this case, we would be choosing $9$ pencils from a set of $10$, which is written as ${\textcolor{w h i t e}{X}}_{10} {C}_{9}$, or $\left(\begin{matrix}10 \\ 9\end{matrix}\right)$.

In general terms, a combination is defined as:

color(white)X_nC_r = (n!)/(r!(n-r)!)

Substituting our values in, we get

color(white)X_10C_9 = (10!)/(9!(10-9)!)

=(10!)/(9! * 1!)

=(10*9!)/(9!*1)

=(10*cancel(9!))/(cancel(9!)*1)

$= 10$

Thus, there are $10$ possible ways to choose a set of $9$ pencils from a selection of $10$.