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

$- {}^{15}{C_1} + 2.{}^{15}{C_2} - 3.{}^{15}{C_3} + ....\,. - 15.{}^{15}{C_{15}}$<br><br>$= \sum\limits_{r = 1}^{15} {{{( - 1)}^r}.r.{}^{15}{C_r}}$<br><br>$= \sum\limits_{r = 1}^{15} {{{( - 1)}^2}.r.{{15} \over r}} .{}^{14}{C_{r - 1}}$<br><br>$= 15\sum\limits_{r = 1}^{15} {{{( - 1)}^2}.{}^{14}{C_{r - 1}}}$<br><br>$$ = 15\left( { - {}^{14}{C_0} + {}^{14}{C_1} - {}^{14}{C_2} + .... - {}^{14}{C_{14}}} \right)$$<br><br>$= 15(0) = 0$<br><br>We know,<br><br>$\Rightarrow$2^{14 - 1}$= {}^{14}{C_1} + {}^{14}{C_3} + {}^{14}{C_5} .... + {}^{14}{C_{13}}$ <br><br>$\Rightarrow$2¹³$= {}^{14}{C_1} + {}^{14}{C_3} + {}^{14}{C_5} .... + {}^{14}{C_{13}}$ <br><br>Also let, S = ¹⁴C₁ + ¹⁴C₃ + ¹⁴C₅ + ...+ ¹⁴C₁₁ <br><br>$\Rightarrow$ S + ¹⁴C₁₃ = ¹⁴C₁ + ¹⁴C₃ + ¹⁴C₅ + ...+ ¹⁴C₁₁ + ¹⁴C₁₃ <br><br>$\Rightarrow$ S + ¹⁴C₁₃ = 2¹³ <br><br>$\Rightarrow$ S + 14 = 2¹³ <br><br>$\Rightarrow$ S = 2¹³ - 14 <br><br><b>Other Method :</b><br><br>We know, $${(1 - x)^{15}} = {}^{15}{C_0} - {}^{15}{C_1}x + {}^{15}{C_2}{x^2} - ..... - {}^{15}{C_{15}}{x^{15}}$$<br><br>Differentiating both sides with respect to x, <br><br>$$15{(1 - x)^{14}}( - 1) = - {}^{15}{C_1} + 2{}^{15}{C_2}x - 3{}^{15}{C_3}{x^2} + ....... - 15{}^{15}{C_{15}}{x^{14}}$$<br><br>Put $x = 1$<br><br>$$ \Rightarrow 0 = - {}^{15}{C_1} + 2{}^{15}{C_2} - 3{}^{15}{C_3} + .... - 15{}^{15}{C_{15}}$$ <br><br>We know,<br><br>$\Rightarrow$2^{14 - 1}$= {}^{14}{C_1} + {}^{14}{C_3} + {}^{14}{C_5} .... + {}^{14}{C_{13}}$ <br><br>$\Rightarrow$2¹³$= {}^{14}{C_1} + {}^{14}{C_3} + {}^{14}{C_5} .... + {}^{14}{C_{13}}$ <br><br>Also let, S = ¹⁴C₁ + ¹⁴C₃ + ¹⁴C₅ + ...+ ¹⁴C₁₁ <br><br>$\Rightarrow$ S + ¹⁴C₁₃ = ¹⁴C₁ + ¹⁴C₃ + ¹⁴C₅ + ...+ ¹⁴C₁₁ + ¹⁴C₁₃ <br><br>$\Rightarrow$ S + ¹⁴C₁₃ = 2¹³ <br><br>$\Rightarrow$ S + 14 = 2¹³ <br><br>$\Rightarrow$ S = 2¹³ - 14