10. Simplify each expression and write it without using factorial notation. (3 marks each) (n+4)! (n+2)!...

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10. Simplify each expression and write it without using factorial notation. (3 marks each) (n+4)! (n+2)! a. b. (n-r+1)! (n-y-

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Answer 1

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$\\ a) \ \frac{(n+4)!}{(n+2)!}=\frac{(n+4)(n+3)(n+2)!}{(n+2)!}=(n+4)(n+3)=n^2+7n+12\\\\\\ b)\ \frac{(n-r+1)}{(n-r-2)!}=\frac{(n-r+1)(n-r)(n-r-1)(n-r-2)!}{(n-r-2)!}\\\\\ =(n-r+1)(n-r)(n-r-1)\\ =(n-r+1)(n-r-1)(n-r)\\ =((n-r)^2-1^2)(n-r)\\ =(n-r)^3-(n-r)\\ =n^3-r^3-3n^2r+3nr^2-n+r$

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10. Simplify each expression and write it without using factorial notation. (3 marks each) (n+4)! (n+2)!...

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