Question

Find out the number of combinations and the number of From Permutations for 9 objects taken 7 at a time.Express your answer in simplest form

Answer 1

The formula for Combination is,

`nCr=(n!)/(r!\left(n-r\right)!)`

Given

`n=9,r=7`

Therefore,

`^9C_7=(9!)/(7!\left(9-7\right)!)`

Simplify

`\begin{gathered} (9!)/(7!\left(9-7\right)!)=(9!)/(7!(2!))=(9*8*7!)/(7!2*1)=(9*8)/(2)=9*4=36 \\ \therefore^9C_7=36 \end{gathered}`

The formula for Permutation is,

`nPr=(n!)/(\left(n-r\right)!)`

Given:

`n=9,r=7`

Therefore,

`^9P_7=(9!)/(\left(9-7\right)!)`

Simplify

`\begin{gathered} (9!)/(\left(9-7\right)!)=(9!)/(2!)=(9*8*7*6*5*4*3*2!)/(2!)=9*8*7*6*5*4*3 \\ =181440 \\ \therefore^9P_7=181440 \end{gathered}`

Hence, there are 36 combinations and 181440 permutations.