Permutation and CombinationHard

Question

The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is

Options

A.429
B.384
C.403
D.455

Solution

Let oranges are identical then

$$x_{1} + x_{2} + x_{3} + x_{4} = 16 $$and $x_{1},x_{2},x_{3},x_{4} \geq 1$

or $x_{1}' + x_{2}' + x_{3}' + x_{4}' = 12$

so total number of solutions are

$$= \ ^{12 + 3}C_{3} = \ ^{15}C_{3} = 455$$

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