+2653 Miyu is giving out $8$ identical chocolates to her $5$ friends, including Dhruv. All possible distributions are equally likely. What is the probability that Dhruv gets at least $6$ chocolates?
+9673 Suppose the friends are A, B, C, Dhruv (D), E. Then denote the number of chocolate gotten by each person as \(x_A, x_B, \cdots, x_E\).
The problem becomes to count the nonnegative integer solutions of \(x_A+x_B+x_C+x_D+x_E=8\) with \(x_D \geq 6\).
We make the substitution \(y_A=x_A, y_B=x_B, y_C=x_C, y_D=x_D-6,y_E=x_E\). Then the equation becomes \(y_A+y_B+y_C+y_D+y_E=2\). We will count the nonnegative integer solutions of this equation to get the answer. The answer is \(\displaystyle \binom{2 + 5 - 1}{5 - 1} = 15\).
