Question:
The value of the expression \(^{47}C_4+ \displaystyle \sum^5_{j = 1} , ^{52-j}C_3\) is
The value of the expression \(^{47}C_4+ \displaystyle \sum^5_{j = 1} , ^{52-j}C_3\) is
Updated On: Aug 29, 2023
- $^{47}C_5$
- $^{52}C_5$
- $^{52}C_4$
- None of these
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The Correct Option is C
Solution and Explanation
The correct answer is C:\(^{52}C_4\)
Given expression;
\(^{47}C_4+ \displaystyle \sum^5_{j = 1} , ^{52-j}C_3\)
\(= ^{47}C_4+ ^{51}C_3 + ^{50}C_3 + ^{49}C_3+ ^{48}C_3 + ^{47}C_3\)
\(= ( ^{47}C_4+ ^{47}C_3 )+ ^{48}C_3 + ^{49}C_3+ ^{50}C_3 + ^{51}C_3\)
\(20mm [using ^nC_r + ^nC_{r-1} = ^{n+1}C_r]\)
\(= ( ^{48}C_4+ ^{48}C_3 )+ ^{49}C_3 + ^{50}C_3 + ^{51}C_3\)
\(= ( ^{49}C_4+ ^{49}C_3 )+ ^{50}C_3 + ^{51}C_3\)
\(= ( ^{50}C_4+ ^{50}C_3 )+ ^{51}C_3\)
\(= ^{51}C_4+ ^{51}C_3 = ^{52}C_4\)
Given expression;
\(^{47}C_4+ \displaystyle \sum^5_{j = 1} , ^{52-j}C_3\)
\(= ^{47}C_4+ ^{51}C_3 + ^{50}C_3 + ^{49}C_3+ ^{48}C_3 + ^{47}C_3\)
\(= ( ^{47}C_4+ ^{47}C_3 )+ ^{48}C_3 + ^{49}C_3+ ^{50}C_3 + ^{51}C_3\)
\(20mm [using ^nC_r + ^nC_{r-1} = ^{n+1}C_r]\)
\(= ( ^{48}C_4+ ^{48}C_3 )+ ^{49}C_3 + ^{50}C_3 + ^{51}C_3\)
\(= ( ^{49}C_4+ ^{49}C_3 )+ ^{50}C_3 + ^{51}C_3\)
\(= ( ^{50}C_4+ ^{50}C_3 )+ ^{51}C_3\)
\(= ^{51}C_4+ ^{51}C_3 = ^{52}C_4\)
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