The number of ways in which 16 identical oran | vedclass

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(D) Let the number of oranges given to the four children be $x_1, x_2, x_3, x_4$ respectively.Since the oranges are identical,we need to find the numb

Vedclass · Accepted Answer

$455$

Vedclass · Answer

(D) Let the number of oranges given to the four children be $x_1, x_2, x_3, x_4$ respectively.Since the oranges are identical,we need to find the number of positive integer solutions to the equation:$x_1 + x_2 + x_3 + x_4 = 16$,where $x_i \geq 1$ for $i = 1, 2, 3, 4$.Using the substitution $x_i = x_i^{\prime} + 1$,where $x_i^{\prime} \geq 0$,the equation becomes:$(x_1^{\prime} + 1) + (x_2^{\prime} + 1) + (x_3^{\prime} + 1) + (x_4^{\prime} + 1) = 16$$x_1^{\prime} + x_2^{\prime} + x_3^{\prime} + x_4^{\prime} = 12$.The number of non-negative integer solutions is given by the formula $\binom{n+k-1}{k-1}$,where $n = 12$ and $k = 4$.Number of ways $= \binom{12+4-1}{4-1} = \binom{15}{3}$.$\binom{15}{3} = \frac{15 	imes 14 	imes 13}{3 	imes 2 	imes 1} = 5 	imes 7 	imes 13 = 455$.

The number of ways in which $16$ identical oranges can be distributed among $4$ children such that each child gets at least one orange is:

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