Question

Sabendo que n é a solução da equação (n-3)! = 24, determine o valor An,2 + Cn,3

Answer 1

$(n-3)!=24\\(n-3)!=4!\\n-3=4\\n=4+3\\n=7$
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$A_{n,2}+C_{n,3}=\dfrac{7!}{(7-2)!}+\dfrac{7!}{3!(7-3)!}\\\\\\A_{n,2}+C_{n,3}=\dfrac{7!}{5!}+\dfrac{7!}{3!4!}\\\\\\A_{n,2}+C_{n,3}=\dfrac{7\cdot6\cdot5!}{5!}+\dfrac{7\cdot6\cdot5\cdot4!}{3\cdot2\cdot1\cdot4!}\\\\\\A_{n,2}+C_{n,3}=42+35\\\\\\\boxed{\boxed{A_{n,2}+C_{n,3}=77}}$