Permutations and Combinations Solutions

• Counting problems that are more complex can be categorized into    permutations    and    combinations   .
• The formulas use the variables n and r.
  • n =    the total number of elements to choose from   
  • r =    the size of the group (of elements)   
• A permutation is an ordered arrangement in which no element is used more than once, and    the arrangement of elements matters   .
• The formula for permutation is: $nPr \left(n, r\right)=\frac{n!}{\left(n–r\right)!}$
• A combination is an arrangement in which no element is used more than once, and    the arrangement of elements makes no difference   .
• The formula for combination is: $nCr \left(n, r\right)=\frac{n!}{r!\left(n–r\right)!}$

Example 4

A teacher has a class of 24 students and plans to divide them into groups of 4. One option is to assign each member of a group a specific role, and the other is to have them work in committees. How many different groups can be formed, with and without specific roles?

Groups with specific roles $nPr \left(n, r\right)=\frac{n!}{\left(n–r\right)!}$ $\begin{array}{rcl}nPr \left(24, 4\right)& =& \frac{24!}{\left(24–4\right)!}\\ & =& \frac{24!}{20!}=\frac{24·23·22·21·\cancel{20!}}{\cancel{20!}}\\ nPr \left(24, 4\right)& =& 255,024\end{array}$ When students are assigned specific roles, there are    255,024    possible ways to arrange the 24 students into groups of 4.

Committees $nCr \left(n, r\right)=\frac{n!}{r!\left(n–r\right)!}$ $\begin{array}{rcl}nCr \left(24, 4\right)& =& \frac{24!}{4!\left(24–4\right)!}\\ & =& \frac{24!}{4!\left(20!\right)}=\frac{{}^6\cancel{24}·23·{}^{11}\cancel{22}·{}^7\cancel{21}·\cancel{20!}}{\left(\cancel{4}·\cancel{3}·\cancel{2}·1\right)\left(\cancel{20!}\right)}\\ & =& 6·23·11·7\\ nCr \left(24, 4\right)& =& 10,626\end{array}$ There are    10,626    different groups of 4 when 24 students are arranged into committees.

Note

Remember to use a calculator for these problems. The number of options is often much greater than you might expect.

Example 5

An investment group was picking stocks from the Fortune 100 list. They plan to have 25 different stocks in their portfolio. How many different options are possible?

$\begin{array}{rcl}n& =& 100, r=25\\ nCr \left(100, 25\right)& =& \frac{100!}{25!\left(100–25\right)!}\\ & =& \frac{100!}{25! 75!}\\ nCr \left(100, 25\right)& =& 2.43\times {10}^{23}\\ & & \end{array}$

There are $\begin{array}{rcl}& & 2.43\times {10}^{23}\end{array}$ ways to select 25 stocks from a list of 100.

Example 6

Javier is creating a playlist of his top ten songs. Determine the number of ways Javier’s songs can be arranged.

$\begin{array}{rcl}nPr \left(10, 10\right)& =& \frac{10!}{\left(10–10\right)!}\\ & =& \frac{10!}{0!}=10!\\ nPr \left(10, 10\right)& =& 3,628,800\end{array}$

Javier can make 3,628,800 different playlists with his top ten songs.