The value of -15C1 + 2.15C2 -- 3.15C3 + ...

Question

The value of
-15C1 + 2.15C2 -- 3.15C3 + ... - 15.15C15 + 14C1 + 14C3 + 14C5 + ...+ 14C11 is :

Accepted Answer

$$ - {}^{15}{C_1} + 2.{}^{15}{C_2} - 3.{}^{15}{C_3} + ....\,. - 15.{}^{15}{C_{15}}$$

$$ = \sum\limits_{r = 1}^{15} {{{( - 1)}^r}.r.{}^{15}{C_r}} $$

$$ = \sum\limits_{r = 1}^{15} {{{( - 1)}^2}.r.{{15} \over r}} .{}^{14}{C_{r - 1}}$$

$$ = 15\sum\limits_{r = 1}^{15} {{{( - 1)}^2}.{}^{14}{C_{r - 1}}} $$

$$ = 15\left( { - {}^{14}{C_0} + {}^{14}{C_1} - {}^{14}{C_2} + .... - {}^{14}{C_{14}}} \right)$$

$= 15(0) = 0$

We know,

$$ \Rightarrow $$214 - 1$$ =  {}^{14}{C_1} + {}^{14}{C_3} + {}^{14}{C_5} .... + {}^{14}{C_{13}}$$

$$ \Rightarrow $$213$$ =  {}^{14}{C_1} + {}^{14}{C_3} + {}^{14}{C_5} .... + {}^{14}{C_{13}}$$

Also let, S = 14C1 + 14C3 + 14C5 + ...+ 14C11

$$ \Rightarrow $$ S  + 14C13 = 14C1 + 14C3 + 14C5 + ...+ 14C11 + 14C13

$$ \Rightarrow $$ S  + 14C13 = 213

$$ \Rightarrow $$ S + 14 = 213

$$ \Rightarrow $$ S = 213 - 14

Other Method :

We know, $${(1 - x)^{15}} = {}^{15}{C_0} - {}^{15}{C_1}x + {}^{15}{C_2}{x^2} - ..... - {}^{15}{C_{15}}{x^{15}}$$

Differentiating both sides with respect to x,

$$15{(1 - x)^{14}}( - 1) =  - {}^{15}{C_1} + 2{}^{15}{C_2}x - 3{}^{15}{C_3}{x^2} + ....... - 15{}^{15}{C_{15}}{x^{14}}$$

Put $x = 1$

$$ \Rightarrow 0 =  - {}^{15}{C_1} + 2{}^{15}{C_2} - 3{}^{15}{C_3} + .... - 15{}^{15}{C_{15}}$$