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Question
Evaluate
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Solution
Given:
\[{}^{20} C_5 + \sum^5_{r = 2} 25 -^r C_4\]
\[ = {}^{20} C_5 +^{23} C_4 + {}^{22} C_4 + {}^{21} C_4 + {}^{20} C_4 \]
\[ = \left( {}^{20} C_{4_{}} + {}^{20} C_5 \right) +^{21} C_4 +^{22} C_4 +^{23} C_4 \]
\[=^{21} C_5 +^{21} C_4 + {}^{{}^{22}} C_4 +^{23} C_4\]
[∵\[{{}^{}}^n C_{r - 1} +^n C_r =^{n + 1} C_r\]]
\[= \left( {}^{21} C_{4_{}} + {}^{21} C_{5_{}} \right) +^{22} C_4 +^{23} C_4 \]
\[ =^{24} C_5\] [∵ \[{{}^{}}^n C_{r - 1} +^n C_r =^{n + 1} C_r\]]
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