If (1+x)¹⁵=a₀+a₁ x+ +a₁₅ x¹⁵, then _ r=1 ¹⁵ r a_r a_r-1 is equal to

Step-by-step solution

Given that aligned &(1+x)¹⁵=a₀+a₁ x+a₂ x^2+ +a₁₅ x¹⁵ & ¹⁵ C₀+ ¹⁵ C₁ x+ ¹⁵ C₂ x^2+ + 15 15 x¹⁵ &=a₀+a₁ x+a₂ x^2+ +a₁₅ x¹⁵ aligned Equating the coefficient of various powers of x , we get aligned & a₀= ¹⁵ C₀, a₁= ¹⁵ C₁, a₂= ¹⁵ C₂, , a₁₅= 15 15 & _ r=1 ¹⁵ r a_r a_ r-1 = _ r=1 ¹⁵ r ¹⁵ C_r 15 r-1 = & _ r=1 ¹⁵ r 15 ! r !(15-r) ! 15 ! (r-1) !(15-r+1) ! = & _ r=1 ¹⁵ (r-1) !(15-r+1) ! r !(15-r) ! = & _ r=1 ¹⁵ 15-r+1=15+14+13+ +2+1 = & 15(15+1) 2 =120 aligned