The value of (300)(3010) - (301)(3011)
+ (302) (3012) + ..............+ (3020) (3030)
(1−x)30 = 30C0x0 - 30C1x1 + 30C2x2
+ .............+ (−1)30 30C30x30 .............(i)
(x+1)30 = 30C0x30 - 30C1x29 + 30C2x28
+ .............+ 30C10x20 + ..........+ 30C30x0 ............(ii)
Multiplying (i) and (ii) and equating the
coefficient of x20 on both sides, we get required
sum = coefficient of x20 in (1−x2)30 = 30C10.