If ^nC_r denotes the number of combination of n things taken r at a time, then the express…

Question

If ${}^n{C_r}$ denotes the number of combination 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

Accepted Answer

Arrange it this way,

$$^n{C_{r + 1}} + 2.{}^n{C_r} + {}^n{C_{r - 1}}$$

$$ = {}^n{C_{r + 1}} + {}^n{C_r} + {}^n{C_r} + {}^n{C_{r - 1}}$$

$$\left[ \, \right.$$ Now use the rule,

$$\left. {{}^n{C_r} + {}^n{C_{r - 1}} = {}^{n + 1}{C_r}} \right]$$

$= {}^{n + 1}{C_{r + 1}} + {}^{n + 1}Cr$

$= {}^{n + 2}{C_{r + 1}}$

Solutions · 1