Question

For $2\le r\le n,(\frac{n}{r})+2(\frac{n}{r-1})+(\frac{n}{r+2})=$

• A. $(\frac{n+1}{r-1})$
• B. $2(\frac{n+1}{r-1})$
• C. $2(\frac{n+2}{r})$
• D. $(\frac{n+2}{r})$

Answer 1

The correct option is A $(\frac{n+1}{r-1})$

${}^n{C}_r+2{}^n{C}_{r-1}{+}^n{C}_{r-2}$ $(2\le r\le n)$

${}^n{C}_r{+}^n{C}_{r-1}+{}^n{C}_{r-1}{+}^n{C}_{r-2}$

${}^{n+1}{C}_r{+}^{n+1}{C}_{r-1}$

${}^{n{+}_2}{C}_r=\frac{(n+2)!}{r!(n+2-r)!}$$=(\frac{n+1}{r-1})$