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QJEEM22Ma7d84: JEE Main 2022 Mathematics - Class 11 JEEM22Ma7d84 Solution: The number of ways to distribute 30 identical candies among…

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 :

Multiple Choice Options

  • A 205
  • B 615
  • C 510
  • D 430

Step-by-Step Verified 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−x4)1−x(1−x5)1−x(1−x31)2(1−x)2{{(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−x4−x5+x9)(1−x)−4(1 - {x^4} - {x^5} + {x^9}){(1 - x)^{ - 4}}

=27C24−23C20−22C19+18C15=430= {}^{27}{C_{24}} - {}^{23}{C_{20}} - {}^{22}{C_{19}} + {}^{18}{C_{15}} = 430

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