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Q.
If $\,{}^nC_r +3 \,{}^nC_{r+ 1} + 3 \,{}^nC_{r+ 2} +\,{}^nC_{r+ 3} = \,{}^{15}C_9$, then which of the following is not true?
Permutations and Combinations
Solution:
Given, $\,{}^nC_r +3 \,{}^nC_{r+ 1} + 3 \,{}^nC_{r+ 2} + \,{}^nC_{r+ 3} = \,{}^{15}C_9$,
As upper suffixes are same, lower suffixes are in increasing order and coefficients of the terms are the coefficients of the expansion of $(1 + x)^3$.
$\therefore \,{}^nC_r + 3\,{}^nC_{r+1} + 3\,{}^nC_{r+2}+ \,{}^nC_{r + 3}$
$ = \,{}^{n + 3}C_{r + 3} = \,{}^{15}C_9 = \,{}^{n + 3}C_{n - r}$
(Using $\,{}^nC_r + \,{}^nC_{r + 1} = \,{}^{n + 1}C_{r +1} $ and $\,{}^nC_r = \,{}^nC_{n - r})$
$\Rightarrow n + 3 = 15 $ and $ r + 3 = 9, r + 3 + 9 = 15$
[By using $\,{}^nC_x = \,{}^nC_y \Rightarrow x + y = n$ or $ x = y]$
$\Rightarrow n = 12, r = 6$ or $3$.