JEE MainMathematicsToughPermutations and Combinations

A man wants to distribute 101 coins of a rupee each; among his
three sons with the condition that no one receives more money that the
combined sum of other two. The number of ways of doing this

Show answer and solution

x, y, z are amounts, then

x \leq y + z \Rightarrow x \leq 101 - x

\Rightarrow x \leq 50

and

y,z \leq 50

x + y + z = 101

\therefore

no. of ways = coefficient
of

x^{101}

\left( 1 + x + x^{2} + \ldots\ldots.. + x^{50} \right)^{3}

= \ ^{103}C_{101} - 3.\ ^{52}C_{50}

Source: Grass-G / M-08 Mathematics Paper 23 October.docx

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