The value of the expression ^47C_4 + sum_j=1^5 ^52-jC_3 is — JEE Main 1980 Mathematics PYQ

1980

The value of the expression

^{47}C_4 + \sum_{j=1}^{5} {}^{52-j}C_3

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

Show answer and solution

Correct option: C

Solution

The sum expands as ⁵¹C₃ + ⁵⁰C₃ + ⁴⁹C₃ + ⁴⁸C₃ + ⁴⁷C₃ plus ⁴⁷C₄.

Use Pascal's identity (nCr + nC(r+1) = (n+1)C(r+1)) repeatedly from the bottom:
⁴⁷C₃ + ⁴⁷C₄ = ⁴⁸C₄, then ⁴⁸C₃ + ⁴⁸C₄ = ⁴⁹C₄, then ⁴⁹C₃ + ⁴⁹C₄ = ⁵⁰C₄, then ⁵⁰C₃ + ⁵⁰C₄ = ⁵¹C₄, then ⁵¹C₃ + ⁵¹C₄ = ⁵²C₄.

[1980]

Practise Permutations and Combinations the way it is asked

A timed set of real JEE Main Permutations and Combinations questions, graded as the exam grades. No sign-up.

Start a Permutations and Combinations test

More: all JEE Main Permutations and Combinations questions · JEE Main 1980 · Mathematics chapter list

JEE Main 1980 Mathematics question on Permutations and Combinations