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Write the Value of 6 ∑ R = 1 56 − R C 3 + 50 C 4 - Mathematics

Write the value of\[\sum^6_{r = 1} \ ^{56 - r}{}{C}_3 + \ ^ {50}{}{C}_4\]

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Solution

We know:
nC\[-\]1 + nCr = n+1Cr

\[\text{Now, we have}: \]
\[ \sum^6_{r = 1} {}^{56 - r} C_3 + {}^{50} C_4 \]
\[ =^{55} C_3 +^{54} C_3 +^{53} C_3 +^{52} C_3 +^{51} C_3 +^{50} C_3 +^{50} C_4\]
\[=^{55} C_3 +^{54} C_3 +^{53} C_3 +^{52} C_3 +^{51} C_3 +^{51} C_4 \]
\[ =^{55} C_3 +^{54} C_3 +^{53} C_3 +^{52} C_3 +^{52} C_4 \]
\[ =^{55} C_3 +^{54} C_3 +^{53} C_3 +^{53} C_4 \]
\[ =^{55} C_3 +^{54} C_3 +^{54} C_4 \]
\[ =^{55} C_3 +^{55} C_4 \]
\[ =^{56} C_4 \]
Concept: Factorial N (N!) Permutations and Combinations
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 17 Combinations
Q 5 | Page 24
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