Question

For the given values of n and r, evaluate (a) n!÷ (n-r)! and (b) n!÷ r!(n-r)! for n=12 and r=5.

Answer 1

a) 95040

b) 792

Step-by-step explanation:

`\begin{gathered} a)\text{ }n!/\mleft(n-r\mright)! \\ =(n!)/(n-r!) \\ \text{where n = 12} \\ r\text{ = 5} \end{gathered}`

Inserting the values into the question above:

`\begin{gathered} =(12!)/(\mleft(12-5\mright)!)\text{ = }(12!)/(7!) \\ =\text{ }\frac{12\text{ }*11\text{ }*10*9*8*7!}{7!} \\ 7!\text{ cancels out in the numerator and denominator} \\ =\text{ }12\text{ }*11\text{ }*10*9*8 \\ =95,040 \end{gathered}`
`\begin{gathered} b)\text{ }n!/ r!\mleft(n-r\mright)! \\ =\text{ }(n!)/(r!\mleft(n-r\mright)!)\text{ } \\ \text{where n = 12, r = 5} \\ =\text{ }(12!)/(7!(12-7)!) \end{gathered}`
`\begin{gathered} =(12!)/(7!(5)!)=(12!)/(7!*5!) \\ =\frac{12*11*10*9*8*7!}{7!\text{ }*5!} \\ =(12*11*10*9*8)/(5*4*3*2*1) \end{gathered}`
`\begin{gathered} =(95040)/(120) \\ =792 \end{gathered}`