`R-C=C-R` reacts with

If C_0, C_1, C_2 ,……..C_n are the coefficient in the expansion of (1 + x)^n then show that C_0 C_r + C_1 C_(r + 1) + C_2 C_(r + 2) + ………..+ C_(n-r).C_n = ((2n)!)/((n-r)!(n+r)!)

If a, b, c are three given non-coplanar vectors and any arbitratry vector r is in space, where Delta_1=|[r*a, b*a, c*a], [r*b, b*b, c*b], [r*c, b*c, c*c]| : Delta_2=|[a*a, r*a, c*a], [a*b, r*b, c*b], [a*c, r*c, c*c]| Delta_3=|[a*a, b*a, r*a], [a*b, b*b, r*b], [a*c, b*c, r*c]|, Delta_4=|[a*a, b*a, c*a], [a*b, b*b, c*b], [a*c, c*c, c*c]| Q. The vector r is expressible in the form

If a, b, c are three given non-coplanar vectors and any arbitratry vector r is in space, where Delta_1=|[r*a, b*a, c*a], [r*b, b*b, c*b], [r*c, b*c, c*c]| : Delta_2=|[a*a, r*a, c*a], [a*b, r*b, c*b], [a*c, r*c, c*c]| Delta_3=|[a*a, b*a, r*a], [a*b, b*b, r*b], [a*c, b*c, r*c]|, Delta_4=|[a*a, b*a, c*a], [a*b, b*b, c*b], [a*c, c*c, c*c]| Q. The vector r is expressible in the form

If a, b, c are three given non-coplanar vectors and any arbitratry vector r is in space, where Delta_1=|[r*a, b*a, c*a], [r*b, b*b, c*b], [r*c, b*c, c*c]| : Delta_2=|[a*a, r*a, c*a], [a*b, r*b, c*b], [a*c, r*c, c*c]| Delta_3=|[a*a, b*a, r*a], [a*b, b*b, r*b], [a*c, b*c, r*c]|, Delta_4=|[a*a, b*a, c*a], [a*b, b*b, c*b], [a*c, c*c, c*c]| Q. The vector r is expressible in the form

a , b , c are three non coplanar non zero vectors and r is any vector in space, then ( a × b )×( r × c )+( b × c )×( r × a )+( c × a )×( r × b ) is equal to) [ ⃗ [2a b ⃗ c ] ⃗ r b. 2 [ ⃗ a ⃗ b ⃗ c ] ⃗ r c. 3 [ ⃗ a ⃗ b ⃗ c ] ⃗ r d. none of these

If C_(r)=10C_(r), then sum_(r=1)^(10)C_(r-1)C_(r) is equal to