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