The number of ways to distribute 30 identical candies among four...

Given, ${C}_{1} + {C}_{2} + {C}_{3} + {C}_{4} = 30$

Now it has given that $4 \le {C}_{2} \le 7 \text{ and } 2 \le {C}_{3} \le 6$

and ${C}_{1} \text{ and } {C}_{4}$ can take any value.

Finding coefficient of ${x}^{30} in</p><p>\left({x}^{0} + {x}^{1} + . . . . . . {x}^{30}\right)\left({x}^{4} + {x}^{5} + . . . . . . {x}^{7}\right)\left({x}^{2} + . . . {x}^{6}\right)\left({x}^{0} + . . . . {x}^{30}\right)$

$= {\left(\dfrac{1 - {x}^{31}}{1 - x}\right)}^{2}{x}^{4}\left(\dfrac{1 - {x}^{4}}{1 - x}\right){x}^{2}\left(\dfrac{1 - {x}^{5}}{1 - x}\right)$

{Ignoring higher power more than $30$}

$= {\left(\dfrac{1}{1 - x}\right)}^{2}{x}^{4}\left(\dfrac{1 - {x}^{4}}{1 - x}\right){x}^{2}\left(\dfrac{1 - {x}^{5}}{1 - x}\right)$

$= {x}^{6}\left(1 - {x}^{4}\right)\left(1 - {x}^{5}\right){\left(1 - x\right)}^{- 4}$ $= {x}^{6}\left(1 - {x}^{5} - {x}^{4} + {x}^{9}\right){\left(1 - x\right)}^{- 4}$

$= \left({x}^{6} - {x}^{11} - {x}^{10} + {x}^{15}\right){\left(1 - x\right)}^{- 4}$

Required coefficient $= C24 4 + 24 - 1 - C19 4 + 19 - 1$ $- C20 20 + 4 - 1 + C15 15 + 4 - 1$

$= C24 27 - C19 22 - C20 23 + C15 18$ $= 430$