First slide

Combinations

Question

$The value of \sum_{r = 0}^{20} ({}^{50 - r}{C_{6}}) is equal to:$

Moderate

A

$^{51} C_{7} -^{30} C_{7}$
B

$^{50} C_{6} -^{30} C_{6}$
C

$^{50} C_{7} -^{30} C_{7}$
D

$^{51} C_{7} +^{30} C_{7}$

Solution

$\sum_{r = 0}^{20} {}^{50 - r}{C_{6}} =^{50} C_{6} +^{49} C_{6} + ...... +^{31} C_{6} + {}^{30}{C_{6} +}^{30} C_{7} -^{30} C_{7}$
$=^{51} C_{7} -^{30} C_{7}$

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