The number of ways to distribute 25 identical | vedclass

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(C) Let the number of apples received by the $4$ boys be $x_1, x_2, x_3, x_4$ respectively.We are given the equation $x_1 + x_2 + x_3 + x_4 = 25$,wher

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$^{12}C_3$

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(C) Let the number of apples received by the $4$ boys be $x_1, x_2, x_3, x_4$ respectively.We are given the equation $x_1 + x_2 + x_3 + x_4 = 25$,where $x_i \ge 4$ for $i = 1, 2, 3, 4$.Let $y_i = x_i - 4$,where $y_i \ge 0$.Substituting $x_i = y_i + 4$ into the equation:$(y_1 + 4) + (y_2 + 4) + (y_3 + 4) + (y_4 + 4) = 25$$y_1 + y_2 + y_3 + y_4 + 16 = 25$$y_1 + y_2 + y_3 + y_4 = 9$Using the stars and bars formula,the number of non-negative integer solutions is given by $^{n+r-1}C_{r-1}$,where $n = 9$ and $r = 4$.Number of ways = $^{9+4-1}C_{4-1} = ^{12}C_3$.

The number of ways to distribute $25$ identical apples among $4$ boys,such that each boy receives at least $4$ apples,is

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