The sum of the series 20C0 - 20C1 + 20C2 - 20C3 ...............+ 20C10 is
1220C10
Consider the expansion of (1+x)20
(1+x)20 = 20C0+20C1x+20C2x2..........20C10x10+20C11x11+20C12x12..........20C20x20
Put x = -1
⇒ 20C0 - 20C1 + 20C2 .............+20C9+20C10−20C11+20C12 + ............ + 20C20
= (−20C0−20C1+20C2...........−20C9)+20C10+(−20C11+20C12+..............+20C20)
20C0=20C20,20C1=20C19,.........20C9=20C11
⇒ 20=2×(20C0−20C1+........−20C9)+20C10
+ 20C10=2(20C0−20C1+20C2........20C9)+2×20C10
(Adding 20C10 on both the sides)
⇒ 20C102 = (20C0−20C1+20C2.........20C10)