Question

The value of the expression $^{47}C_4+ \displaystyle \sum^5_{j = 1} , ^{52-j}C_3$ is

• A. $^{47}C_5$
• B. $^{52}C_5$
• C. $^{52}C_4$
• D. None of these

Answer 1

Answer: (C)

$^{52}C_4$

Answer 2

The correct answer is C:$^{52}C_4$
Given expression;
$^{47}C_4+ \displaystyle \sum^5_{j = 1} , ^{52-j}C_3$
$= ^{47}C_4+ ^{51}C_3 + ^{50}C_3 + ^{49}C_3+ ^{48}C_3 + ^{47}C_3$
$= ( ^{47}C_4+ ^{47}C_3 )+ ^{48}C_3 + ^{49}C_3+ ^{50}C_3 + ^{51}C_3$
$20mm [using ^nC_r + ^nC_{r-1} = ^{n+1}C_r]$
$= ( ^{48}C_4+ ^{48}C_3 )+ ^{49}C_3 + ^{50}C_3 + ^{51}C_3$
$= ( ^{49}C_4+ ^{49}C_3 )+ ^{50}C_3 + ^{51}C_3$
$= ( ^{50}C_4+ ^{50}C_3 )+ ^{51}C_3$
$= ^{51}C_4+ ^{51}C_3 = ^{52}C_4$