In how many ways can Rs. 16 be divided among | vedclass

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(B) Let the amounts received by the $4$ persons be $x_1, x_2, x_3, x_4$ such that $x_1 + x_2 + x_3 + x_4 = 16$,where $x_i \ge 3$ for each $i \in \{1,

Vedclass · Accepted Answer

$35$

Vedclass · Answer

(B) Let the amounts received by the $4$ persons be $x_1, x_2, x_3, x_4$ such that $x_1 + x_2 + x_3 + x_4 = 16$,where $x_i \ge 3$ for each $i \in \{1, 2, 3, 4\}$.Let $y_i = x_i - 3$,where $y_i \ge 0$.Substituting $x_i = y_i + 3$ into the equation:$(y_1 + 3) + (y_2 + 3) + (y_3 + 3) + (y_4 + 3) = 16$$y_1 + y_2 + y_3 + y_4 + 12 = 16$$y_1 + y_2 + y_3 + y_4 = 4$Using the stars and bars formula,the number of non-negative integer solutions is given by $\binom{n + k - 1}{k - 1}$,where $n = 4$ and $k = 4$.Number of ways = $\binom{4 + 4 - 1}{4 - 1} = \binom{7}{3} = \frac{7 	imes 6 	imes 5}{3 	imes 2 	imes 1} = 35$.

In how many ways can Rs. $16$ be divided among $4$ persons if none of them gets less than Rs. $3$?

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