Question

If ${}^n{C}_r+3{}^n{C}_{r+1}+3{}^n{C}_{r+2}+{}^n{C}_{r+3}={}^{15}{C}_9$, then which of the following is not true?

• A. $n=12$
• B. $r=6$
• C. $r=3$
• D. None of these

Answer 1

Answer: (D)

Given, ${}^n{C}_r+3{}^n{C}_{r+1}+3{}^n{C}_{r+2}+{}^n{C}_{r+3}={}^{15}{C}_9$,
As upper suffixes are same, lower suffixes are in increasing order and coefficients of the terms are the coefficients of the expansion of $(1+x{)}^3$.
$\therefore {}^n{C}_r+3{}^n{C}_{r+1}+3{}^n{C}_{r+2}+{}^n{C}_{r+3}$
$={}^{n+3}{C}_{r+3}={}^{15}{C}_9={}^{n+3}{C}_{n-r}$
(Using ${}^n{C}_r+{}^n{C}_{r+1}={}^{n+1}{C}_{r+1}$ and ${}^n{C}_r={}^n{C}_{n-r})$
$\Rightarrow n+3=15$ and $r+3=9,r+3+9=15$
[By using ${}^n{C}_x={}^n{C}_y\Rightarrow x+y=n$ or $x=y]$
$\Rightarrow n=12,r=6$ or $3$.