Question

Use the formula for n^P_r to solve the following equation. At the benefit concert, twelve bands have volunteered to perform but there is only enough time for eight of the bands to play. How many lineups are possible?

Answer 1

Step-by-step explanation:

Step 1. Of 20 bands, only 8 will play at the concert. Let n be the total number of bands, and r be the number of bands that will play:

`\begin{gathered} n=20 \\ r=80 \end{gathered}`

Step 2. We are asked to use the nPr formula, which is a formula to calculate the number of permutations when we have n total objects and we take r at a time:

`_nP_r=(n!)/((n-r)!)`

Step 3. Substitute the known values of n and r to find the answer:

`_(20)P_8=(20!)/((20-8)!)`

Simplifying the result:

`_(20)P_8=(20!)/(12!)`

We expand the factorial expressions:

`_(20)P_8=(20*19*18*17*16*15*14*13*12!)/(12!)`

12! on the numerator cancels with 12! on the denominator:

`_(20)P_8=20*19*18*17*16*15*14*13`

The result is:

`_(20)P_8=5,079,110,400`

Answer:

`\begin{equation*} 5,079,110,400 \end{equation*}`