Express the given product or quotient as a factorial expression. Assume that \( n > r > 0 \).

1. \( (n+1)n! \)

2. \(\frac{(n+8)!}{n+8}\)

3. \(100 \times 99 \times 98\)

4. \((n-1)(n-2)(n-3)\)

1.) (n+1)n! = (n+1)!

2.) (n+8)!/n+8 = (n+8)!/(n+8)
= (n+8)*(n+8-1)! / (n+8) = ((n+8)*(n+7)!) / (n+8)
Cancel out the (n+8)
= (n +7)!


3.) 100 x 99 x 98 = (100 x 99 x 98 x 97!) / (97!) = (100!) / (97!).


(n-1)(n-2)(n-3) = ((n-1)(n-2)(n-3) (n-4)!) /(n-4)!
= (n-1)! / (n-4)!.

Cheers. That's it.

Answered by olemakpadu | 2024-06-10

Each of the given expressions can be expressed in factorial form as follows: ( n + 1 ) n ! = ( n + 1 )! , n + 8 ( n + 8 )! ​ = ( n + 7 )! , 100 × 99 × 98 = 97 ! 100 ! ​ , and ( n − 1 ) ( n − 2 ) ( n − 3 ) = ( n − 4 )! ( n − 1 )! ​ .
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Answered by olemakpadu | 2024-09-26