number of ways of distributing 10 identical balls among 3 kids such that each of kid gets atleast one ball

To solve the problem of distributing 10 identical balls among 3 kids with the condition that each kid receives at least one ball, we can start by giving one ball to each kid. This uses up 3 balls, leaving us with 7 balls to distribute. Now, the problem reduces to distributing these 7 identical balls among 3 kids, where some kids can receive zero balls. The formula for distributing \( n \) identical items into \( r \) distinct groups is given by the "stars and bars" theorem. Here, we need to calculate the number of non-negative integer solutions to the equation: \[ x_1 + x_2 + x_3 = 7 \] The number of solutions can be computed using the formula: \[ \binom{n + r - 1}{r - 1} \] In this case, \( n = 7 \) (the remaining balls) and \( r = 3 \) (the kids), so we have: \[ \binom{7 + 3 - 1}{3 - 1} = \binom{9}{2} = 36 \] So, there are 36 ways to distribute the 10 identical balls among 3 kids, ensuring that each kid gets at least one ball! Think of it like hosting a party! You’ve got 10 identical cupcakes to share with 3 kids. After giving each one a cupcake to start, you’re left with 7 cupcakes to distribute however you like. The way to figure out the distribution magic is like placing invisible dividers (the bars) to separate the cupcakes into kid portions, leading to some seriously sweet math skills! This topic touches on combinatorics, which is a branch of mathematics dealing with counting, arrangements, and combinations. If you're keen to dive deeper, consider exploring resources like "Enumerative Combinatorics" by Richard Stanley, which beautifully expands on various methods to count and arrange objects!