Question

JEE MAINS MATHS QUESTION 2020

Answer 1

ANSWER.

$\sf \to \: \sum \limits_{r \: = 0} {}^{20} \: \: \: {}^{50 - r} c_{6} = {}^{51} c_{7} - {}^{30} c_{7}$

$\sf \to \: put \: the \: value \: of \: r \: = 20 \: \: to \: \: r = 0 \: \: \: in \: decreasing \: form \\ \\ \sf \to \: {}^{30}c_{6} + {}^{31} c_{6} + {}^{32}c_{6} + ........ + {}^{50} c_{6} \\ \\ \sf \to \: {}^{n} c_{r} + {}^{n}c_{r + 1} = {}^{n + 1}c_{r + 1} \\ \\ \sf \to \: {}^{n + 1}c_{r + 1} = {}^{31}c_{7}$

$\sf \to \: {}^{30}c_{6} + {}^{31} c_{6} + {}^{32}c_{6} +.... + {}^{50} c_{6} - {}^{30}c_{7} - {}^{31}c_{7} - ..... - {}^{50}c_{7} \\ \\ \sf \to \: {}^{n}c_{r + 1} = {}^{50}c_{7} \\ \\ \sf \to \: {}^{50}c_{7} - {}^{30} c_{7} \\ \\ \sf \to \: \therefore \: last \: term \: is \: also \: increased \: by \:( n + 1 ) \: factor$