What are the number of ways of distributing $15$ fruits each to $2$ people from $5$ apples, $10$ mangoes and $15$ oranges?

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Solution

The correct option is D 66
The desired number is the coefficient of $x^{15}$ in
$(1 + x^{2} + x^{3} . . . x^{5}) (1 + x^{2} + x^{3} . . . x^{10}) (1 + x^{2} + . . . x^{15})$
$= (1 - x^{6}) (1 - x^{11}) (1 - x^{16}) (1 - x)^{- 3}$
Hence, the desired number will be $^{17} C_{15} -^{11} C_{9} -^{6} C_{4}$
$=^{17} C_{2} -^{11} C_{2} -^{6} C_{2}$
$= 136 - 55 - 15$
$= 66$

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What are the number of ways of distributing 15 fruits each to 2 people from 5 apples, 10 mangoes and 15 oranges?

The correct option is D 66The desired number is the coefficient of x15 in(1+x2+x3...x5)(1+x2+x3...x10)(1+x2+...x15)=(1−x6)(1−x11)(1−x16)(1−x)−3Hence, the desired number will be 17C15−11C9−6C4=17C2−11C2−6C2=136−55−15=66

What are the number of ways of distributing $15$ fruits each to $2$ people from $5$ apples, $10$ mangoes and $15$ oranges?