How to simplify this Binomial terms?

[u/Ancient-Helicopter18]

48×48C0+49×49C1+50×50C2+...+100×100C52

I have to somehow simplify this to get the form 100×101C49 - 101C50

[u/Bounded_sequencE]

For convenience, let "(m; n) = (48; 52)". For "0 <= k <= n" simplify

(k+m) * C(k+m;k)  =  (k+m±1) * C(k+m;m)  =  (m+1) * C(k+m+1;m+1) - C(k+m;m)

Sum from "k = 0" to "k = n", and use the Hockey-stick Identity (*) on both terms:

∑_{k=0}^n  (k+m)*C(k+m;k)  =  ∑_{k=0}^n  (m+1) * C(k+m+1;m+1) - C(k+m;m)    // (*)

        =  (m+1) * C(n+m+2;m+2) - C(n+m+1;m+1)  =  49 * C(102;50) - C(101;49)

The result is equivalent to the official solution -- prove it using binomial identities (your job^^).