The value of $\sum\limits_{r = 0}^{20} {{}^{50 - r}{C_6}}$ is equal to:
Solution
$$\sum\limits_{r = 0}^{20} {} {}^{50 - r}{C_6} = {}^{50}{C_6} + {}^{49}{C_6} + {}^{48}{C_6} + .... + {}^{30}{C_6}$$<br><br>$$ = {}^{50}{C_6} + {}^{49}{C_6} + .... + {}^{31}{C_6} + ({}^{30}{C_6} + {}^{30}{C_7}) - {}^{30}{C_7}$$<br><br>$$ = {}^{50}{C_6} + {}^{49}{C_6} + .... + ({}^{31}{C_6} + {}^{31}{C_7}) - {}^{30}{C_7}$$<br><br>$= {}^{50}{C_6} + {}^{50}{C_7} - {}^{30}{C_7}$<br><br>$= {}^{51}{C_7} - {}^{30}{C_7}$<br><br>[As ${{}^n{C_r} + {}^n{C_{r - 1}} = {}^{n + 1}{C_r}}$]
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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