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Question Number 49982 by pieroo last updated on 12/Dec/18

If (n_P_r  /n_C_r  ) −1=5, find the value of r

$$\mathrm{If}\:\frac{\mathrm{n}_{\mathrm{P}_{\mathrm{r}} } }{\mathrm{n}_{\mathrm{C}_{\mathrm{r}} } }\:−\mathrm{1}=\mathrm{5},\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{r} \\ $$

Answered by ajfour last updated on 13/Dec/18

^n P_r  = ((n!)/((n−r)!))   &   ^n C_r  = ((n!)/(r!(n−r)!))  ⇒   ((^n P_r  )/(^n C_r ))−1 = r!−1 = 5  ⇒   r! = 6   ⇒   r= 3 .

$$\:^{{n}} {P}_{{r}} \:=\:\frac{{n}!}{\left({n}−{r}\right)!}\:\:\:\&\:\:\:\:^{{n}} {C}_{{r}} \:=\:\frac{{n}!}{{r}!\left({n}−{r}\right)!} \\ $$$$\Rightarrow\:\:\:\frac{\:^{{n}} {P}_{{r}} \:}{\:^{{n}} {C}_{{r}} }−\mathrm{1}\:=\:{r}!−\mathrm{1}\:=\:\mathrm{5} \\ $$$$\Rightarrow\:\:\:{r}!\:=\:\mathrm{6}\:\:\:\Rightarrow\:\:\:{r}=\:\mathrm{3}\:. \\ $$

Commented by ajfour last updated on 13/Dec/18

2! = 2  ,  4! = 24  ⇒  2! <r! < 4!  lets check 3! = 1×2×3 = 6  ⇒  r! = 6  ⇒  r=3 .

$$\mathrm{2}!\:=\:\mathrm{2}\:\:,\:\:\mathrm{4}!\:=\:\mathrm{24} \\ $$$$\Rightarrow\:\:\mathrm{2}!\:<{r}!\:<\:\mathrm{4}! \\ $$$${lets}\:{check}\:\mathrm{3}!\:=\:\mathrm{1}×\mathrm{2}×\mathrm{3}\:=\:\mathrm{6} \\ $$$$\Rightarrow\:\:{r}!\:=\:\mathrm{6}\:\:\Rightarrow\:\:{r}=\mathrm{3}\:. \\ $$

Commented by pieroo last updated on 13/Dec/18

thanks sir, But how do i explain to   my younger brother that r!=6 ⇒r=3?

$$\mathrm{thanks}\:\mathrm{sir},\:\mathrm{But}\:\mathrm{how}\:\mathrm{do}\:\mathrm{i}\:\mathrm{explain}\:\mathrm{to}\: \\ $$$$\mathrm{my}\:\mathrm{younger}\:\mathrm{brother}\:\mathrm{that}\:\mathrm{r}!=\mathrm{6}\:\Rightarrow\mathrm{r}=\mathrm{3}? \\ $$

Commented by pieroo last updated on 13/Dec/18

wow, thank you very much for your patience.

$$\mathrm{wow},\:\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{for}\:\mathrm{your}\:\mathrm{patience}. \\ $$

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