^{47}{C₄} + mathop Σ limits_{r = 1}^5 {}^{52 - r}{C₃} =

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6.Permutation and Combination

medium

$^{47}{C_4} + \mathop \sum \limits_{r = 1}^5 {}^{52 - r}{C_3} = $

A

$^{47}{C_6}$

B

$^{52}{C_5}$

C

$^{52}{C_4}$

D

None of these

(IIT-1980)

Solution

(c) $^{47}{C_4} + \sum\limits_{r = 1}^5 {^{52 – r}{C_3}} { = ^{51}}{C_3}{ + ^{50}}{C_3}{ + ^{49}}{C_3}{ + ^{48}}{C_3}{ + ^{47}}{C_3}{ + ^{47}}{C_4}$

${ = ^{51}}{C_3}{ + ^{50}}{C_3}{ + ^{49}}{C_3}{ + ^{48}}{C_3}{ + ^{48}}{C_4}$

${ = ^{51}}{C_3}{ + ^{50}}{C_3}{ + ^{49}}{C_3}{ + ^{49}}{C_4}$

${ = ^{51}}{C_3}{ + ^{50}}{C_3}{ + ^{50}}{C_4}{ + ^{51}}{C_3}{ + ^{51}}{C_4}$

$={^{52}}{C_4}$.

Standard 11

Mathematics

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