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Question
Write the value of\[\sum^6_{r = 1} \ ^{56 - r}{}{C}_3 + \ ^ {50}{}{C}_4\]
Solution
We know:
nCr \[-\]1 + nCr = n+1Cr
\[ \sum^6_{r = 1} {}^{56 - r} C_3 + {}^{50} C_4 \]
\[ =^{55} C_3 +^{54} C_3 +^{53} C_3 +^{52} C_3 +^{51} C_3 +^{50} C_3 +^{50} C_4\]
\[ =^{55} C_3 +^{54} C_3 +^{53} C_3 +^{52} C_3 +^{52} C_4 \]
\[ =^{55} C_3 +^{54} C_3 +^{53} C_3 +^{53} C_4 \]
\[ =^{55} C_3 +^{54} C_3 +^{54} C_4 \]
\[ =^{55} C_3 +^{55} C_4 \]
\[ =^{56} C_4 \]
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