Please solve this answer is option A.............................

S= ∑r*⁴⁰C_r*³⁰C_r from r=0 to 30

now, we use the formula: r*ⁿC_r= n*^n-1C_r-1 (note that this formula is extremely easy to prove)

so, S= ∑40³⁹C_r-1*³⁰C_r from r=0 to 30

S= 40*∑³⁹C_r-1​*³⁰C_r from r=1 to 30

= 40*∑³⁹C_k*³⁰C_k+1 from k=0 to 29

now, ∑³⁹C_29-k*³⁰C_k= coefficient of x^29 in (30C0*x^0+30C1*x^1+....+30C30*x^30)*(39C0*x^0+39C1*x^1+.....+39C39*x^39)

= coefficient of x^29 in (1+x)^30*(1+x)^39= (1+x)^69

= 69C29

hence, S= 40*⁶⁹C₂₉

note that the above ques is quite hard and if you need some more solved examples of these types, kindly refer TMH advanced book and you will find plenty of them along with complete solutions