Checkout JEE MAINS 2022 Question Paper Analysis : Checkout JEE MAINS 2022 Question Paper Analysis :

The value of
\({ }^{47} \mathrm{C}_{4}+\sum_{j=1}^{5} 52-\mathrm{j} \mathrm{C}_{3}\)
is equal to?

1) 47C5

2) 52C5

3) 52C4

4) 53C7

Answer: (3) 52C4

Solution:

\(\begin{array}{l}{ }^{47} \mathrm{C}_{4}+\sum_{j=1}^{5} 52-\mathrm{j} \mathrm{C}_{3}\end{array} \)

= 47C4 + 51C3+ 50C3+49C3+48C3+47C3

= 47C4+47C3+ 51C3+50C3+49C3+48C3

= 48C4+48C3+49C3+50C3+51C3 {Since, nCr + nCr-1 = n+1Cr}

= 49C4+49C3+50C3+51C3

= 50C4+50C3+51C3

= 51C4+51C3

= 52C4

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