Question

If ${}^n{C}_r$ denots the number of combinations of $n$ things taken $r$ at a time, then the expression ${}^n{C}_{r+1}{+}^n{C}_{r-1}+2{\times }^n{C}_r$ equals

• A. ${}^{n+2}{C}_r$
• B. ${}^{n+2}{C}_{r+1}$
• C. ${}^{n+1}{C}_r$
• D. ${}^{n+1}{C}_{r+1}$

Answer 1

Answer: (B)

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

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

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

${=}^{n+1}{\text{C}}_{r+1}{+}^{n+1}{\text{C}}_r$

${=}^{n+2}{\text{C}}_{r+1}$