The number of ways to distribute 30 identical candies among four children C₁, C₂, C₃ and C₄ so that C₂ receives at least 4 and at most 7 candies, C₃ receives at least 2 and at most 6 candies, is equal to :

<p>By multinomial theorem, no. of ways to distribute 30 identical candies among four children C₁, C₂ and C₃, C₄</p> <p>= Coefficient of x³⁰ in (x⁴ + x⁵ + .... + x⁷) (x² + x³ + .... + x⁶) (1 + x + x² ....)²</p> <p>= Coefficient of x²⁴ in $${{(1 - {x^4})} \over {1 - x}}{{(1 - {x^5})} \over {1 - x}}{{{{(1 - {x^{31}})}^2}} \over {{{(1 - x)}^2}}}$$</p> <p>= Coefficient of x²⁴ in $(1 - {x^4} - {x^5} + {x^9}){(1 - x)^{ - 4}}$</p> <p>$= {}^{27}{C_{24}} - {}^{23}{C_{20}} - {}^{22}{C_{19}} + {}^{18}{C_{15}} = 430$</p>