Please solve all question with each and every step mentioned only then I will upvote:

Example 13. Find the coefficient of x50 in the polynomials (i) (1 + x)1000 + x(1 + x)999 + x2(1 + x)998 + .... + x 1000 (ii) (1+ x)+2(1 +x)2+3(1+x)3+....+ 1000(1 +x) 1000Example 14. Find the coefficient in x in the expansion of the expression ( 1 + x ) k + (1 + x)k+]+ ...... + (1 + x)" in powers of x. Consider the cases m k.Example 16. Find the summation of the following series "C2+2. n+IC,+3. n+2C+ . . . + 2n-1C32n 2n Example 18. If Ea, (x -2)' = _b, (x -3)" and r=0 r=0 a = 1 for all k > n, then show that b. = 2+ C nExample 19. Prove that (i) C +2C, + 3C,+ ....+.C = n.20-1 ii) Co+ 2C, + 3C, + ....+ (n+ 1)C =(n+2) 20-1